The End Of Christianity – Finding a Good God in an Evil World – Pg.31
William Dembski PhD. Mathematics
Excerpt: “In mathematics there are two ways to go to infinity. One is to grow large without measure. The other is to form a fraction in which the denominator goes to zero. The Cross is a path of humility in which the infinite God becomes finite and then contracts to zero, only to resurrect and thereby unite a finite humanity within a newfound infinity.” http://www.designinference.com.....of_xty.pdf

Of note: I hold ‘growing large without measure’ to be a lesser quality infinity than the infinity in which ‘a fraction in which the denominator goes to zero’. The main principle for why I hold growing large without measure to be a ‘lesser quality infinity’ is stated at the 4:30 minute mark of the following video:

Can A “Beginning-less Universe” Exist? – William Lane Craig – video ,,”the impossiblity of forming an actually infinite number of things by adding one member after another.,,,
1. A collection formed by adding one member to another cannot be actually infinite,,,” http://www.youtube.com/watch?v=K8YN0fwo5J4

It is also interesting to note that the conflict of reconciling General Relativity and Quantum Mechanics arises from the inability of either theory to successfully deal with the Zero/Infinity conflict that crops up in different places of each theory:

THE MYSTERIOUS ZERO/INFINITY
Excerpt: The biggest challenge to today’s physicists is how to reconcile general relativity and quantum mechanics. However, these two pillars of modern science were bound to be incompatible. “The universe of general relativity is a smooth rubber sheet. It is continuous and flowing, never sharp, never pointy. Quantum mechanics, on the other hand, describes a jerky and discontinuous universe. What the two theories have in common – and what they clash over – is zero.”,, “The infinite zero of a black hole — mass crammed into zero space, curving space infinitely — punches a hole in the smooth rubber sheet. The equations of general relativity cannot deal with the sharpness of zero. In a black hole, space and time are meaningless.”,, “Quantum mechanics has a similar problem, a problem related to the zero-point energy. The laws of quantum mechanics treat particles such as the electron as points; that is, they take up no space at all. The electron is a zero-dimensional object,,, According to the rules of quantum mechanics, the zero-dimensional electron has infinite mass and infinite charge. http://www.fmbr.org/editoral/e....._mar02.htm

Moreover there is actual physical evidence that lends strong support to the position that the ‘Zero/Infinity conflict’, we find between General Relativity and Quantum Mechanics, was successfully dealt with by Christ:

The Center Of The Universe Is Life (Jesus) – General Relativity, Quantum Mechanics, Entropy and The Shroud Of Turin – video http://vimeo.com/34084462

Moreover, as would be expected if General Relativity, Quantum Mechanics/Special Relativity (QED) were truly unified in the resurrection of Christ from death, the image on the shroud is found to be formed by a quantum process. The image was not formed by a ‘classical’ process:

The absorbed energy in the Shroud body image formation appears as contributed by discrete values – Giovanni Fazio, Giuseppe Mandaglio – 2008
Excerpt: This result means that the optical density distribution,, can not be attributed at the absorbed energy described in the framework of the classical physics model. It is, in fact, necessary to hypothesize a absorption by discrete values of the energy where the ‘quantum’ is equal to the one necessary to yellow one fibril. http://cab.unime.it/journals/i.....802004/271

As well, as would be expected in such a ‘singularity’ reconciling Gravity with Quantum Mechanics, Gravity appears to have been overcome in the resurrection event of Christ:

Particle Radiation from the Body – July 2012 – M. Antonacci, A. C. Lind
Excerpt: The Shroud’s frontal and dorsal body images are encoded with the same amount of intensity, independent of any pressure or weight from the body. The bottom part of the cloth (containing the dorsal image) would have born all the weight of the man’s supine body, yet the dorsal image is not encoded with a greater amount of intensity than the frontal image. Radiation coming from the body would not only explain this feature, but also the left/right and light/dark reversals found on the cloth’s frontal and dorsal body images. https://docs.google.com/document/d/19tGkwrdg6cu5mH-RmlKxHv5KPMOL49qEU8MLGL6ojHU/edit

Verses and Music:

Philippians 2: 6-9
Who, though he was in the form of God, did not regard equality with God something to be grasped.
Rather, he emptied himself, taking the form of a slave, coming in human likeness; and found human in appearance, he humbled himself, becoming obedient to death, even death on a cross.
Because of this, God greatly exalted him and bestowed on him the name that is above every name,

Colossians 1:19-20
For God was pleased to have all his fullness dwell in him, and through him to reconcile to himself all things, whether things on earth or things in heaven, by making peace through his blood, shed on the cross.

0 is the sum of all positive and negative numbers. It is unique in that it is neither positive nor negative. 0 is the loneliest number. This is why everything comes from nothing and amounts to nothing. It is the reason that we live in a yin-yang or symmetric universe. The ultimate law of physics is the conservation of nothing. Motion is caused by the universe correcting a violation to the conservation of nothing.

0 is why the universe is ONE. 🙂

To conclude, what did the Zen master say to the hot dog vendor? Answer: Make me one with everything. 😀

Get Joe to watch this one (from BA77?s post), he doesn’t believe in Cantor’s mathematics.

Actually the fact is I have proven that his one-to-one correspondence is contrived,rather than derived, with respect to infinite sets in which one set is a proper subset of the other. That means his one-to-one correspondence is not what it appears as obviously there isn’t one and only one match between the two sets.

Leave it to Jerad to not be able to grasp that simple fact.

Infinity is crackpottery for a surprisingly simple reason.

Compared to infinity any finite quantity is infinitely small.

That’s it. In other words, if one accepts infinity, one must accept a reality where quantities are both finite and infinitely small at the same time. Alternatively, infinity cannot exist because it cannot be compared to any finite quantity without introducing a logical contradiction.

It gets worse, much worse. Without infinity, continuous structures, which are infinitely smooth by definition, cannot exist. Not long before his death, Einstein wrote to his friend Besso:

I consider it quite possible that physics cannot be based on the field concept, i. e., on continuous structures. In that case nothing remains of my entire castle in the air gravitation theory included, [and of] the rest of modern physics. – Einstein in a 1954 letter to Besso, quoted from: Subtle is the Lord, Abraham Pais.

In other words, physics is sitting on a mountain of crap.

Mapou: Compared to infinity any finite quantity is infinitely small. That’s it. In other words, if one accepts infinity, one must accept a reality where quantities are both finite and infinitely small at the same time.

Yikes! No!

Your first statement concerns the ratio of a finite quantity to infinity. Your second statement conflates the finite quantity (which is finite) with the ratio (which is an infinitesimal).

alter-M: Compared to 1000, 1 is only 1/1000th. That’s it. In other words, if one accepts 1000, one must accept a reality where quantities are both 1 and 1/1000th at the same time.

Mapou: Compared to infinity any finite quantity is infinitely small. That’s it. In other words, if one accepts infinity, one must accept a reality where quantities are both finite and infinitely small at the same time.

Yikes! No!

Is this reaction supposed to intimidate me or what? Let’s take a look at your logic, if we can call it that.

Your first statement concerns the ratio of a finite quantity to infinity. Your second statement conflates the finite quantity (which is finite) with the ratio (which is an infinitesimal).

I do no such thing.

alter-M: Compared to 1000, 1 is only 1/1000th. That’s it. In other words, if one accepts 1000, one must accept a reality where quantities are both 1 and 1/1000th at the same time.

1/1000 is not a comparison; it’s just a number. A comparison is not a number. It is a test that returns a Boolean value, either true or false. In other words, I am asking two questions:

1. Is 1 infinitely smaller than infinity, yes or no?
2. Is 1 finite, yes or no?

If both answers are affirmative, then there is a contradiction.

PS. I realize that Darwinists and materialists are weavers of lies and deception but you take the cake, Zacky. You are stupider than I thought. But since you love to refer to yourself with the plural “we”, it follows that you are all stupid.

1. The ratio of 1 to infinity is an infinitesimal.
2. 1 is finite.

There is no contradiction.

Amazing. Of course there is a contradiction and it is a blatant one. Does it not follow that, if the ratio of X to infinity is an infinitesimal, that X is infinitely smaller than infinity? And is it not true that infinity is infinitely greater than any finite value X? If true on both counts, then infinity is crackpottery.

You don’t get it, Zachriel. When you compare two quantities, X and Y, you are not asking for a ratio. You are looking for a truth value. Heck, forget ratios. This is not the way comparisons are done in the first place. A CPU does not use ratios to compare two values. It just subtracts one value from the other. The result can be negative, positive or zero. A CPU cannot even compare infinity to another number because its registers are finite. This alone refutes infinity. We know infinity is greater than a finite number, not because we can perform the comparison operation but because it is true by definition.

So your entire ratio argument is bogus right off the bat.

Mapou: When you compare two quantities, X and Y, you are not asking for a ratio. You are looking for a truth value.

You need to be specific. Do you mean X is less than Y. Sure. But you wanted a comparative, that is, how much smaller.

Mapou: It just subtracts one value from the other. The result can be negative, positive or zero.

So you don’t want a truth value, nor the ratio, but the difference. That’s fine. The difference is infinity.

Mapou: 1. Is 1 infinitely smaller than infinity, yes or no?
2. Is 1 finite, yes or no?

1. The difference of 1 and infinity is infinity.
2. 1 is finite.

Still no contradiction. By the way, and there are special rules for the mathematics of infinity.

Nonsense. There are three possible comparison operations that can be performed on X and Y: greater than, less than and equal. Each returns a Boolean value, not a number. Wake up.

Bzzzt. Nobody is saying that 1 is not finite. 1 is not just smaller than infinity. It is infinitely smaller than infinity while being finite at the same time. You cannot say this about any other comparison. That’s the qualitative difference that you insist on ignoring.

Cantor, do you need a rigorous mathematical definition in order to understand that infinity is infinitely greater than any finite number?

And do you need a rigorous mathematical definition to understand that the inverse (any finite number is infinitely smaller than infinity) is also true?

Mapou: do you need a rigorous mathematical definition in order to understand that infinity is infinitely greater than any finite number?

It’s your claim that you can provide a mathematical proof that “infinity is crackpottery”, even though mathematicians nearly universally accept the concept. That means it’s your responsibility to provide unambiguous definitions of your terms.

It’s your claim that you can provide a mathematical proof that “infinity is crackpottery”, even though mathematicians nearly universally accept the concept. That means it’s your responsibility to provide unambiguous definitions of your terms.

I agree completely but I’m not sure why you insist on stating that most Christians believe in infinity.
William Lane Craig (who many consider a leading Christian spokesman) to my knowledge shares your opinion: https://www.youtube.com/watch?v=4X6XKKGo5GY

It’s your claim that you can provide a mathematical proof that “infinity is crackpottery”, even though mathematicians nearly universally accept the concept. That means it’s your responsibility to provide unambiguous definitions of your terms.

Mathematicians can kiss my asteroid because they have a lame pony in this race. They’re already on the record for claiming that infinity exists and they will look bad if the opposite is shown to be true. Mathematicians are political/religious animals just like Darwinists, therefore they are not to be trusted.

My peers are the public. If a concept cannot be explained in simple terms that the average intelligent layperson can understand, it’s crap, IMO.

I agree completely but I’m not sure why you insist on stating that most Christians believe in infinity.
William Lane Craig (who many consider a leading Christian spokesman) to my knowledge shares your opinion: https://www.youtube.com/watch?v=4X6XKKGo5GY

Thanks for the link. Actually, most Christians, Catholics and Protestants, believe that God is infinitely knowledgeable and powerful. They call it omniscience and omnipotence. It’s heresy, IMO, the work of the devil 😀 . I say this because it introduces all sorts of logical and ethical problems.

It is the only number that is neither positive nor negative. It is the only number, when divided by itself does not equal one.
It is the only divisor that produces an undefined result.
Repeated divisions by two still produce an even number.

There is an even simpler proof that infinity is bogus: If a quantity cannot be counted by any possible computer, that quantity cannot exist.

I place the concept of infinity in the same category as Darwinism: a gigantic fraud perpetrated on the public for nefarious reasons. Some evil forces are trying extremely hard to keep humanity from acquiring certain forbidden knowledge about the universe. In particular, we are being kept from understanding the following:

1 The universe is necessarily discrete and absolute, and had a beginning.
2. Space (distance) is but a perceptual illusion.
3. A time dimension is pure crackpottery.

I’m glad you appreciate my ideas, cantor. It fills me with great warmth. Here’s a little question that’s bothering me. Maybe you can help. I mean, an accomplished and brilliant mathematician like you should have no trouble knowing the right answer. A simple yes or no will suffice. I assure you it is not a trick.

Can a body move in spacetime?

PS. Zachriel is also welcome to take a shot at it.

There is an even simpler proof that infinity is bogus: If a quantity cannot be counted by any possible computer, that quantity cannot exist.

Whether you believe in infinity or not, don’t tell me computers can’t “count” infinity. Every computer can calculate infinity. Heard of Basel problem ? Try calculating the sum. Your computer will do it in a jiffy. The answer is Pi^2/6

You do realize that numbers aren’t actually real things, right?

What do you think the largest prime number is? Since numbers are just concepts, there is no largest number. Infinity is nether the largest nor the smallest number, since it doesn’t represent anything.

Mapou: Mathematicians can kiss my asteroid because they have a lame pony in this race. They’re already on the record for claiming that infinity exists and they will look bad if the opposite is shown to be true.

Infinity is an abstraction, just like -1 or a hyperbolic paraboloid. Your claim is that the concept is inconsistent. To show that requires a mathematical proof.

Mapou: My peers are the public. If a concept cannot be explained in simple terms that the average intelligent layperson can understand, it’s crap, IMO.

Like quantum mechanics or general relativity, or for that matter, the circuits in your CPU.

dgw: Zero is an odd number.

Heh.

cantor: Both arguments make a lot of sense… if you don’t think about them.

Ha!

Mapou: There is an even simpler proof that infinity is bogus: If a quantity cannot be counted by any possible computer, that quantity cannot exist.

Argument by redefinition! Tee hee!

ETA: By that definition, there is no decimal expansion of 1/3.

phoodoo: You do realize that numbers aren’t actually real things, right? What do you think the largest prime number is? Since numbers are just concepts, there is no largest number. Infinity is nether the largest nor the smallest number, since it doesn’t represent anything.

So close! You do realize that infinity is a concept, just like numbers are concepts?

alter-phoo: Infinity is nether the largest nor the smallest number, since it is not a number.

Mapou: Mathematicians can kiss my asteroid because they have a lame pony in this race. They’re already on the record for claiming that infinity exists and they will look bad if the opposite is shown to be true.

Infinity is an abstraction, just like -1 or a hyperbolic paraboloid. Your claim is that the concept is inconsistent. To show that requires a mathematical proof.

No, it requires simple logical proof that anybody can understand. I’m claiming that the concept has nothing to do with reality because it cannot be computed by any possible computer that we can think of. Therefore it’s crap.

Mapou: My peers are the public. If a concept cannot be explained in simple terms that the average intelligent layperson can understand, it’s crap, IMO.

Like quantum mechanics or general relativity, or for that matter, the circuits in your CPU.

There is no reason that the circuits of a CPU cannot be explained to a layperson in a language that they can understand. Don’t bring QM into the discussion since physicists have no clue what is going on either.

PS. I’m still waiting for you and your infinity buddy, cantor, to answer the question I posed earlier:

Can a body move in spacetime?

This should not be so hard since you guys are so smart, right? After all, did not Einstein teach that bodies move along their worldlines or geodesics? Stop being a wussy.

Mapou: No, it requires simple logical proof that anybody can understand.

A simple proof is preferred, however, any valid proof would be acceptable.

Mapou: I’m claiming that the concept has nothing to do with reality because it cannot be computed by any possible computer that we can think of.

Actually, computers can be programmed to calculate infinities, just like they can be programmed to calculate using integers or imaginary numbers. They can’t count to infinity, however, because they are finite machines, which, in case you missed it, means not infinite.

Mapou: Don’t bring QM into the discussion since physicists have no clue what is going on either.

So, according to the Mapou Principle, quantum mechanics is “crap” because it can’t be explained to an intelligent layperson.

Whether you believe in infinity or not, don’t tell me computers can’t “count” infinity. Every computer can calculate infinity. Heard of Basel problem ? Try calculating the sum. Your computer will do it in a jiffy. The answer is Pi^2/6

Mapou: It’s a logical proof that infinity does not exist in the universe.

Infinity is an abstract concept, like -1 or a geometric line. All mathematics is abstraction.

Mapou: No infinite series exists because it cannot be computed.

You seem to keep conflating countable with computable. In any case, in your medieval mathematics, there is no decimal expansion of 1/3. Not sure your regressive mathematics will catch on.

Mapou: All kinds of series can go on ad infinitum.

You just used the word infinitum, meaning infinity.

I now fully realize that I’m arguing with willing morons.

Zacky @34:

Mapou: It’s a logical proof that infinity does not exist in the universe.

Infinity is an abstract concept, like -1 or a geometric line. All mathematics is abstraction.

You do realize that abstractions do not exist, right?

Mapou: No infinite series exists because it cannot be computed.

You seem to keep conflating countable with computable. In any case, in your medieval mathematics, there is no decimal expansion of 1/3. Not sure your regressive mathematics will catch on.

I have no problem with expanding 1/3. Maybe you do.

Mapou: All kinds of series can go on ad infinitum.

You just used the word infinitum, meaning infinity.

Wow. I can also talk about angels dancing on the head of a pin.

Give it up, Zacky. You got your foot in your mouth.

Mapou: You do realize that abstractions do not exist, right?

That’s right. Abstractions don’t exist as physical realities. So 3+2i, a conic section, and -1, are abstractions, which may or may not represent some aspect of reality. Infinity is an abstraction just like the number 2. http://www.youtube.com/watch?v=E-kMsBMh6Ng

Mapou: I have no problem with expanding 1/3.

Nor do we! In decimal, it’s infinite and repeating, but not in ternary.

Speaking of crackpottery, belief that the left and right hemispheres of the brain represent yin and yang is crackpottery. So is the believe that everything came from nothing.

But once You believe one crackpot idea why not believe them all?

Perhaps infinity is a left brain concept and very real and not infinity is a right brain concept and very unreal.

I don’t understand why the majority of ID-supporting UD contributors are ignorant of the work done bey Georg Cantor in mathematics which is now an accepted part of mathematics.

There are different ‘sizes’ of infinity. The smallest one is aleph-naught, the ‘size’ of the natural counting numbers. Also referred to as countably infinite.

Any set which can be lined up, element per element, with the natural counting numbers is also said to be countably infinite. This lining up is called a one-to-one correspondence. IT DOES NOT refer to the value 1 but one element in one set matched with one and only one element in the second set.

Countably infinite sets include the natural counting numbers, the positive even integers, the positive odd numbers, the primes, the rational numbers (nice proof that), etc.

The real numbers are a ‘larger size’ of infinity. It is not possible to put them into a one-to-one correspondence with the natural counting numbers. And there’s a very nice proof of that!!

If we add an even number to an even number we get an even number. If we had an even number to an odd number we get an odd number. And if we add an odd number to an even number we get an odd number. And if we add an odd number to an odd number we get an even number.

Let X = any undisputed odd number
Let Y = any undisputed even number

I don’t understand why the majority of ID-supporting UD contributors are ignorant of the work done bey Georg Cantor in mathematics which is now an accepted part of mathematics.

Accepted but apparently not very useful. For example what difference would it be to say that a set of infinite numbers is larger than any of its also infinite proper subsets?

That is we keep the natural one-to-one correspondence used to determine one is a proper subset of the other and the cardinality is determined by a function that maps across positioning in the set.

Mapou
“I think Zacky suffers from some form of autism. I’ve had enough. Adios.”

LOL, A guy named Charles Martel from another forum which Zachy frequented as a time waster said that they figured he suffered from Asperger Syndrome. They banned him over there and someone said he came to this forum and sure enough, his M.O. hasn’t changed one iota.

Accepted but apparently not very useful. For example what difference would it be to say that a set of infinite numbers is larger than any of its also infinite proper subsets?

Not very useful in a practical, everyday situation I grant you. In fact, practically useless. But, at the time, mathematics was in the throes of trying to make sure its foundations were rock solid. and there were certain concepts, like limits, that hadn’t been firmly established theoretically.

Cantor’s work was EXTREMELY controversial at the time. Many mathematicians refused to accept it. But now, it’s considered by most to be a building block of mathematics.

That is we keep the natural one-to-one correspondence used to determine one is a proper subset of the other and the cardinality is determined by a function that maps across positioning in the set.

When you’re dealing with finite sets things are always easier. 🙂 Also, even with finite sets there are usually many ways to make a one-to-one correspondence between two sets.

Here’s an infinite set, let’s call it Z+ = {1, 2, 3, 4, 5 . . . .}

Here’s another infinite set E = {2, 4, 6, 8, 10 . . . . }

You can make the correspondence in this way:

Z+ –> E

1 –> 2
2 –> 4
3 –> 6
etc

This correspondence (or rule or function) is nice because we can even state a simple rule that describes it:

C: the rule that maps an element,n, of Z+ to 2n in E

But we could have made the following correspondence:

1 –> 4
2 –> 2
3 –> 8
4 –> 6
etc

We still get a one-to-one correspondence (i.e. every element of Z+ gets matched to one and only one element of £), it’s harder to define but no element in either set gets left behind.

Also, sometimes you deal with infinite sets that are not easy to ‘order’. For example, consider the rational numbers, Q, made up of all ratios of integers. So 1, 2, 3, 4 . . . are in there as are 1/2, 1/3, 1/4, 1/5 . . . and 2/3, 2/5, (2/4 or 1/2 already appears), 2/7 are in there as are 3/2, 3/4, 3/5 . . .

Finding a way to assign (or contrive if you like) a one-to-one correspondence between Q and Z+ is much tricker although it is possible. In fact, finding such a correspondence is part of one proof that the real numbers have a larger cardinality than the natural counting numbers, Z+.

Countably infinite sets include the natural counting numbers, the positive even integers, the positive odd numbers, the primes, the rational numbers (nice proof that), etc.

The problem is not that infinity is not a viable mathematical concept. The problem is that scientists and others believe that infinite sets exist in nature. This is why we have crackpot nonsense like continuous structures, nonsense that even Einstein, Mr. Continuity par excellence, was beginning to doubt. An infinite number of points on a line and infinitely smooth surfaces are pure unmitigated hogwash in the not even wrong category.

The reason that continuity is still considered a viable concept in physics is that physicists are political (i.e., gutless) animals and nobody can say anything bad against mathematicians and retain their careers. Once you remove infinity from physics, all the infinity mathematicians become obsolete. The jackasses have retarded progress in science by centuries.

I’m still waiting for the resident materialist/Darwinist know-it-alls to gather enough huevos to answer the simple little challenge I posed. Einstein claimed that bodies moved in spacetime along their worldlines or geodesics. Answer the following question with a yes or a no. You can either agree with Einstein or disagree.

The problem is not that infinity is not a viable mathematical concept. The problem is that scientists and others believe that infinite sets exist in nature. This is why we have crackpot nonsense like continuous structures, nonsense that even Einstein, Mr. Continuity par excellence, was beginning to doubt. An infinite number of points on a line and infinitely smooth surfaces are pure unmitigated hogwash in the not even wrong category.

Well, I don’t know many objects or things in life that are infinite BUT some of the models we use are. For example, a sine wave extends out to infinity. The normal curves we use for probability distributions extend out to infinity (I know, the data doesn’t but the mathematical model, which is useful, does). Likewise the force of gravity doesn’t just end at some point, it extends out to infinity even if there is a practical limit. The solutions to differential equations that are used to model electrical systems extend out to infinity.

On the quantum scale entanglement doesn’t seem to be limited to some distance. And black holes are something like a singularity, i.e. what you get when you divide by zero.

The reason that continuity is still considered a viable concept in physics is that physicists are political (i.e., gutless) animals and nobody can say anything bad against mathematicians and retain their careers. Once you remove infinity from physics, all the infinity mathematicians become obsolete. The jackasses have retarded progress in science by centuries.

I don’t think that’s true at all. In fact, in my experience, the relationship between mathematics and physics can be quite tense. And we’ve made the greatest breakthroughs in physics since Cantor did his work. I don’t think continuity has harmed physics at all.

Mapou #68

I’m still waiting for the resident materialist/Darwinist know-it-alls to gather enough huevos to answer the simple little challenge I posed. Einstein claimed that bodies moved in spacetime along their worldlines or geodesics. Answer the following question with a yes or a no. You can either agree with Einstein or disagree.

Well, I don’t know many objects or things in life that are infinite BUT some of the models we use are. For example, a sine wave extends out to infinity. The normal curves we use for probability distributions extend out to infinity (I know, the data doesn’t but the mathematical model, which is useful, does). Likewise the force of gravity doesn’t just end at some point, it extends out to infinity even if there is a practical limit. The solutions to differential equations that are used to model electrical systems extend out to infinity.

Nobody knows where the forces of gravity or electric fields end. Also, gravity breaks down at very small distances where the inverse square law predicts infinite gravity (at r = 0). If this were true, all particles would collapse on themselves. This is not observed.

On the quantum scale entanglement doesn’t seem to be limited to some distance. And black holes are something like a singularity, i.e. what you get when you divide by zero.

This is obviously crackpottery of the worst kind since dividing by zero is nonsense.

Mapou: The reason that continuity is still considered a viable concept in physics is that physicists are political (i.e., gutless) animals and nobody can say anything bad against mathematicians and retain their careers. Once you remove infinity from physics, all the infinity mathematicians become obsolete. The jackasses have retarded progress in science by centuries.

I don’t think that’s true at all. In fact, in my experience, the relationship between mathematics and physics can be quite tense. And we’ve made the greatest breakthroughs in physics since Cantor did his work. I don’t think continuity has harmed physics at all.

That’s funny because I have not seen anything, let alone breakthroughs, derived from the assumption that nature uses either infinity or infinitesimals. Even the use of calculus, the so-called math of continuous structures which is presented as an example of infinity in practice, does not use infinity at all. In fact, I routinely solve calculus functions on my computer and I can assure you, my computer is as finite and discrete as can be.

Mapou #68

I’m still waiting for the resident materialist/Darwinist know-it-alls to gather enough huevos to answer the simple little challenge I posed. Einstein claimed that bodies moved in spacetime along their worldlines or geodesics. Answer the following question with a yes or a no. You can either agree with Einstein or disagree.

Can a body move in spacetime?

Don’t they alway move in spacetime?

Maybe I’m confused.

You are not just confused; you have been deceived and taken to the cleaners. There can be no motion in spacetime whatsoever. This is why Karl Popper called spacetime, called a block universe in which nothing happens. However, I’ll delay my explanation of why there can be no motion in spacetime until somebody in the materialist/Darwinist camp can gather up the guts to offer a refutation of my position.

“phoodoo: You do realize that numbers aren’t actually real things, right? What do you think the largest prime number is? Since numbers are just concepts, there is no largest number. Infinity is nether the largest nor the smallest number, since it doesn’t represent anything.

So close! You do realize that infinity is a concept, just like numbers are concepts?

alter-phoo: Infinity is nether the largest nor the smallest number, since it is not a number.”

Do you think there is some contradiction there? There most certainly isn’t. Lots of things that are concepts are not numbers. Big is a concept (its not a number!) The beginning is a concept. The end is a concept. Fruitless is a concept. Infinity is a concept, it is not a number. It can’t even be used in mathematics, because every answer it gives is nonsense. It is the same as saying 3 X Big=?

The equation is nonsense. Infinity is not a number.

Nobody knows where the forces of gravity or electric fields end. Also, gravity breaks down at very small distances where the inverse square law predicts infinite gravity (at r = 0). If this were true, all particles would collapse on themselves. This is not observed.

I don’t think the extent of the force of gravity or electric fields end at all. They just get weaker and weaker.

The inverse-square law for gravity always refers to the distance between the centres of the masses so r is never actually zero.

This is obviously crackpottery of the worst kind since dividing by zero is nonsense.

It’s analogous, not an exact comparison.

That’s funny because I have not seen anything, let alone breakthroughs, derived from the assumption that nature uses either infinity or infinitesimals. Even the use of calculus, the so-called math of continuous structures which is presented as an example of infinity in practice, does not use infinity at all. In fact, I routinely solve calculus functions on my computer and I can assure you, my computer is as finite and discrete as can be.

Well, as I said, the models used in physics extend out infinitely far. It’s true that there are lots of perfectly good finite numerical methods which can give you very, very good approximate results but

arcsin(1) = pi/2 +/- 2pi x n where n = 1, 2, 3, 4 . . . .

Not only are there an infinite number of exact solutions but part of the exact answers is pi which has an infinite, non-repeating decimal expansion. Any decimal expression used for calculations is just an approximation. When doing things like Fourier analysis we try and find exact answers which then can be approximated as needed.

Fourier analysis involves finding infinite sequences representations for sets of data. It’s used all the time in some areas of engineering.

Remember that many commonly used quantities (like pi and the square root of 2 and e) have infinitely long decimal expansions so you do use infinite things all the time!!

You are not just confused; you have been deceived and taken to the cleaners. There can be no motion in spacetime whatsoever. This is why Karl Popper called spacetime, called a block universe in which nothing happens. However, I’ll delay my explanation of why there can be no motion in spacetime until somebody in the materialist/Darwinist camp can gather up the guts to offer a refutation of my position.

I’m not volunteering to refute your position, whatever it is. But surely all things that exist move through spacetime. My atoms are moving through time certainly and, owing to the motions of the earth, solar system, galaxy and universe through space. But, like I said, maybe I’m confused.

phoodoo #71

It can’t even be used in mathematics, because every answer it gives is nonsense. It is the same as saying 3 X Big=?

Except it is used in mathematics all the time. You can take the limit of a quantity to infinity and it’s quite common to take definite integrals out to infinity (which is defined to be a limit really). You add up infinitely large sequences all the time.

Here’s an infinite set, let’s call it Z+ = {1, 2, 3, 4, 5 . . . .}

Here’s another infinite set E = {2, 4, 6, 8, 10 . . . . }

Great. We use the derived relationship to show one is a proper subset of the other. And we should use the same relationship for everything else. To use a contrived relationship is bogus, but you don’t seem to be able to grasp that simple fact.

If we take set Z+ and subtract set E from it, we get another infinite set- the set of all positive odd integers. That alone proves that set Z+ and set E do not have the same number of elements.

Jerad, good at following, not so good at thinking for himself.

Joe @ 74 – now label all of the positive odd integers (i.e. O=Z+\E) with their ranks. Aren’t their ranks just Z+? If not, which element(s) of Z+ aren’t a rank of O?

Yes, Jerad, I already know that you are very limited. Here try this: contrived:

: having an unnatural or false appearance or quality

It means the same in math as it does to the rest of the world.

That’s perfectly good definition of contrived. BUT contrived has no specific mathematical definition. So if you say: that correspondence is contrived it only means that it seems a bit false. But it doesn’t mean it’s not valid!!

You have yet to show that contrived has been adapted for a specific mathematical meaning. And so . . .

Your complaints about contrived correspondences are trivial.

Despite several requests you have failed to provide a mathematically specific definition of contrived. Which means you cannot MATHEMATICALLY object to this or that correspondence because there is no mathematical definition of contrived.

You painted yourself into the corner Joe. And then you failed to find the path out.

More than a bit false and yes it is validly false.

Whatever that means. In Joe world. And he’s not going to tell us.

I know no ID supporter on UD will criticise another supporter. Maybe it’s because you feel beset upon from all sides. But, seriously, if you can’t as a group decide what you do and don’t believe in then those on the outside don’t know whether to take someone like Joe seriously

You are not just confused; you have been deceived and taken to the cleaners. There can be no motion in spacetime whatsoever. This is why Karl Popper called spacetime, called a block universe in which nothing happens. However, I’ll delay my explanation of why there can be no motion in spacetime until somebody in the materialist/Darwinist camp can gather up the guts to offer a refutation of my position.

I’m not volunteering to refute your position, whatever it is. But surely all things that exist move through spacetime. My atoms are moving through time certainly and, owing to the motions of the earth, solar system, galaxy and universe through space. But, like I said, maybe I’m confused.

Well you are. Nothing can move through time or spacetime by definition. This little inconvenient truth is understood by a few people in the physics community but you will not see it mentioned much anywhere because it makes a lot of the claims regarding Einstein’s spacetime physics look rather silly. I was waiting for some of my more strident detractors to take the bait but bravado is not synonymous with gonads in the materialist/Darwinist community. Here’s the simple reason that nothing can move in spacetime, which also why there can be no time dimension. The following is copied from a blog article I wrote more than four years ago titled Why Einstein’s Physics Is Crap 😀 :

Why Is There No Time Dimension?

The short answer is that a time dimension would make motion impossible because a changing time coordinate is self-referential. The slightly longer answer is that motion in time assumes a velocity in time which would have to be given as v = dt/dt, which is nonsensical. It is that simple, folks. In other words, things like spacetime trajectories, geodesics, objects moving along their world-lines in spacetime are all hogwash. Nothing moves in spacetime, period. Don’t let die-hard relativists pull a wool over your eyes with bullshit non-explanations of why there is motion in spacetime. It’s all crap. This simple truth reveals famous physicists like Albert Einstein, Stephen Hawking (Mr. Black Hole), Michio Kaku (the crackpot on TV), Kip Thorne (Mr. Wormhole), Brian Greene (Mr. String Theory) and others for what they are, a bunch of spacetime crackpots.

The Time Dilation Crap

I am accusing relativists of being a bunch of crackpots and of teaching their crackpottery to generations of students. I am accusing them of putting an effective monkey wrench in the works that prevents the progress of science. Why? Because if there is a time dimension as they claim, there can be no motion. Since motion is observed, there can be no time dimension which means that they are false teachers. The time dimension mindset condemns researchers to chasing after red herrings and prevents them from seeing nature as it is.

Many relativists will, of course, go into an apoplectic fit of rage at my accusations but I don’t care. As a rebel, I find it amusing. Some will inevitably retort that time dilation is proof that time can change or that it is a form of time travel. Don’t you believe any of it. Clock slowing is not due to time dilation but to energy conservation at work. That’s all. Besides, a clock does not measure the passing of time but temporal intervals. If a clock slows down, it follows that the measured intervals will be longer than the previous ones. Time dilation is not just a misnomer, it is a stupid misnomer simply because time cannot change by definition.

Now I realize that most materialists/Darwinists prefer the opinion of authority than an actual simple proof. So I prepared the following:

Here are a few quotes from knowledgeable people regarding the impossibility of motion or change in Einstein’s spacetime:

“There is no dynamics within space-time itself: nothing ever moves therein; nothing happens; nothing changes. […] In particular, one does not think of particles as “moving through” space-time, or as “following along” their world-lines. Rather, particles are just “in” space-time, once and for all, and the world-line represents, all at once the complete life history of the particle.”
Source: Relativity from A to B by Dr. Robert Geroch, U. of Chicago

“According to Einstein’s doctrine the world is a finite four dimensional sphere full with force-lines. No motion is possible in it since time is one of its geometrical dimensions, and there is no external time.”
Source: Methodologia (pdf) by Dr. Uri Fidelman.

“What has been has indeed objectively been and is no more. What will be, objectively is not and has not been (and, in fact, is not even fully determined, according to quantum indeterminacy). All physical systems ride the universal wave of becoming. Any awareness (ours or that of other intelligences) of past and future reflects the objective wave of becoming. There is no problem of “the arrow of time.” There simply is no arrow of time, as if time could go one “way” rather than another. That metaphor is an unfortunate result of spatializing time. The picture of time as a line along which one might travel in one direction or the other is a conceptual disaster. Time is becoming. Becoming is change. The undoing of a change is also a change. There is no “unbecoming.”
Source: “Time, c, and nonlocality: A glimpse beneath the surface?” Physics Essays, vol. 7, pp. 335-340, 1994 by Professor Joe Rosen

“At the same time I realized that such myths may be developed, and become testable; that historically speaking all — or very nearly all — scientific theories originate from myths, and that a myth may contain important anticipations of scientific theories. Examples are Empedocles’ theory of evolution by trial and error, or Parmenides’ myth of the unchanging block universe in which nothing ever happens and which, if we add another dimension, becomes Einstein’s block universe (in which, too, nothing ever happens, since everything is, four-dimensionally speaking, determined and laid down from the beginning). I thus felt that if a theory is found to be non-scientific, or “metaphysical” (as we might say), it is not thereby found to be unimportant, or insignificant, or “meaningless,” or “nonsensical.” But it cannot claim to be backed by empirical evidence in the scientific sense — although it may easily be, in some genetic sense, the “result of observation.”
Source: Conjectures and Refutations by Karl Popper. Emphasis added.

There is neither space nor time. There is only the changing present. Time travel is for total morons. But don’t tell that to Stephen Hawking and his band of clueless followers because they’ll take offence. Enjoy.

OK UD- How many people think that when we look to see if one set is a proper subset of another that we use relationships that are derived. For example the infinite set {2,4,6,8…} is a proper subset of {1,2,3,4,5,6,…} because all of the elements in the first set are also contained in the second. Subsets are derivatives of the (super)set that contains it. There is a natural match of like numbers.

Does everyone agree with this? If not I am open to correction so please speak up

<blockquote.Despite several requests you have failed to provide a mathematically specific definition of contrived.

Why do you think that is a requirement? The definition of contrived is what it is.

You were using it in a specifically mathematical context, implying that it had a mathematical meaning. You contrasted it with ‘deriived’ but you HAVE NOT been able to provide a quote or context where that MATHEMATICAL distinction is made.

You have not shown how to MATHEMATICALLY distinguish between contrived and derived.

Time to stop dancing and to start proving your case.

You have not shown how to MATHEMATICALLY distinguish between contrived and derived.

Yes, I have. For example the infinite set {2,4,6,8…} is a proper subset of {1,2,3,4,5,6,…} because all of the elements in the first set are also contained in the second. Subsets are derivatives of the (super)set that contains it. There is a natural match of like numbers.

The derived relationship is what we use to determine whether or not one set is a proper subset of another.

All other relationships between the two sets are contrived.

And geez, I have been saying that for months. Time to stop with your willful ignorance.

There is neither space nor time. There is only the changing present. Time travel is for total morons. But don’t tell that to Stephen Hawking and his band of clueless followers because they’ll take offence. Enjoy.

Whatever. I’m not sure I need to respond.

Joe #84

OK UD- How many people think that when we look to see if one set is a proper subset of another that we use relationships that are derived. For example the infinite set {2,4,6,8…} is a proper subset of {1,2,3,4,5,6,…} because all of the elements in the first set are also contained in the second. Subsets are derivatives of the (super)set that contains it. There is a natural match of like numbers.

Does everyone agree with this? If not I am open to correction so please speak up

Clearly {2, 4, 6, 8, 10 . . . } is a subset of {1, 2, 3, 4, . . .} It’s the use of the terms derived and contrived that is in contention.

Joe has asserted (without citations) that derived and contrived are meaningful mathematical terms when working in Set Theory. I say they aren’t. I can’t find mathematical definitions of those terms in regards to Set Theory. I studied a bit of set theory and I don’t remember derived and contrived being important ways of distinguishing types of one-to-one correspondence.

Please be honest and judge this disagreement based on the mathematical literature.

It’s the use of the terms derived and contrived that is in contention.

Only by people ignorant of the language.

Please be honest and judge this disagreement based on the mathematical literature.

LoL! Please be honest and judge this disagreement based on the commonly used definitions of the words and the context they are being used.

I say there is one derived relationship between two sets and we use it to determine whether or not on set is a proper subset of another. And all other relationships are contrived. I have made my case in comments 84 & 86.

That is and has always been my argument regardless of how Jerad wants to try to spin it.

Yes, I have. For example the infinite set {2,4,6,8…} is a proper subset of {1,2,3,4,5,6,…} because all of the elements in the first set are also contained in the second. Subsets are derivatives of the (super)set that contains it. There is a natural match of like numbers.

Subsets are’derivatives’ of the set that contains it?

The derived relationship is what we use to determine whether or not one set is a proper subset of another.

JOE, just provide a proper reference for your usage. You’ve been asked many times. Time to put up.

All other relationships between the two sets are contrived.

This has NO mathematical meaning.

.And geez, I have been saying that for months. Time to stop with your willful ignorance.

Well, let’s deal with it once and for all. Provide your reference extolling your use of derived and contrived regarding Set Theory.

LoL! Please be honest and judge this disagreement based on the commonly used definitions of the words and the context they are being used.

But that gets to the exact point of the matter. Joe strongly implied that ‘derived’ correspondences’ were preferable to ‘contrived’ correspondences in a strictly mathematical sense. He has been unable to provide that strictly mathematical definition. I looked, I thought I might be wrong. I couldn’t find it.

When you have scientific/mathematical discussions then the terms you use can take on very specific/particular meanings and you have to be aware of that.

‘Wave’ means many different things depending on the context. And if you’re talking to a physicist they you’d best be precise. Most of you won’t know that ‘path’ is a rather specific term in an area of mathematics called Graph Theory. Derive does have a general definition and usage. But that doesn’t mean that when you use it in a mathematical context you can be so lenient. This happens in theology as well as some of you will know. In any discipline it’s necessary to get specific.

Joe is not using the terms ‘derive’ and contrived’ in any clearly defined, set theory specific way. And so they do not necessarily mean what he thinks they mean. And he is not providing any references for his interpretations.

I say there is one derived relationship between two sets and we use it to determine whether or not on set is a proper subset of another. And all other relationships are contrived. I have made my case in comments 84 & 86.

And I say this is nonsensical. But Joe does a good job of standing his ground which I do applaud.

That is and has always been my argument regardless of how Jerad wants to try to spin it.

I continue to asset that the terms ‘derived’ and ‘contrived’ have no specific mathematical meaning in Set Theory one-to-one correspondences.

I say there is one derived relationship between two sets and we use it to determine whether or not on set is a proper subset of another. And all other relationships are contrived. I have made my case in comments 84 & 86.

And I say this is nonsensical.

Yes, I know you are very limited. There is one natural match of numbers between a subset and its superset. That is the derived relationship. Period. And it doesn’t matter if Jerad gets all angry and stomps his feet.

I continue to asset that the terms ‘derived’ and ‘contrived’ have no specific mathematical meaning in Set Theory one-to-one correspondences.

Well I made my case and Jerad refuses to address it. I will leave it for people to judge for themselves.

But we’re talking mathematics. Not the same thing.

Yes, a subset is derived from some other (super)set.

Why not say extracted or culled?

You used a word, you got called on it, and now you’re desperately trying to defend it.

Do you understand the natural match of numbers between a subset and its superset?

I’ve never read/seen a definition of a ‘natural’ match of numbers between a subset and its superset. Please provide one.

And, please note, Joe has STILL not provided a mathimatical context or reference for his use of contrived and derived in a set theory context. Just saying.

Joe strongly implied that ‘derived’ correspondences’ were preferable to ‘contrived’ correspondences in a strictly mathematical sense.

They mean something in science. Biology uses those words. Why is mathematics immune to them? Does Jerad think that there isn’t any natural match of numbers between a subset and a superset? Really?

Here’s an infinite set, let’s call it Z+ = {1, 2, 3, 4, 5 . . . .}

Here’s another infinite set E = {2, 4, 6, 8, 10 . . . . }

Great. We use the derived relationship to show one is a proper subset of the other. And we should use the same relationship for everything else. To use a contrived relationship is bogus, but you don’t seem to be able to grasp that simple fact.

If we take set Z+ and subtract set E from it, we get another infinite set- the set of all positive odd integers. That alone proves that set Z+ and set E do not have the same number of elements.

Jerad, good at following, not so good at thinking for himself.

I say there is one derived relationship between two sets and we use it to determine whether or not on set is a proper subset of another. And all other relationships are contrived. I have made my case in comments 84 & 86.

And yet Joe STILL cannot provide any references to mathematical usages of his terms. Nor has he even tried.

Yes, I know you are very limited. There is one natural match of numbers between a subset and its superset. That is the derived relationship. Period. And it doesn’t matter if Jerad gets all angry and stomps his feet.

But it does matter if you can defend your statements with proper academic references. You haven’t even tried.

Still, Joe has not established that his use of ‘contrived’ and ‘derived’ have any basis in Set Theory. And I’ve never heard of a general ‘natural match’ principle.

Well I made my case and Jerad refuses to address it. I will leave it for people to judge for themselves.

How can I address a ghost? Something with no corporeal grounding.

I implore all those who read this discussion to respond honestly. There is no point in having and sponsoring discussions if we’re not honest.

AND it would be nice if y’all didn’t just assume I’m a troll and I’m lying. It would be nice to be treated as a human being. Okay?

They mean something in science. Biology uses those words. Why is mathematics immune to them? Does Jerad think that there isn’t any natural match of numbers between a subset and a superset? Really?

I’ve asked you many times to provide your set theory specific meaning of those terms. You haven’t done so. What’s the problem?

No one is saying mathematics is immune but if the terms aren’t strictly defined in context then they have little explanatory power.

Jerad is stuck inside of his little box and his limited intellect won’t allow him to escape.

Nice to see that UD’s policy of open, honest and respectful discussion is being promulgated.

So is the match supernatural, nonnatural, pre natural or artificial? Or do you think your difficulty with the language means something?

You can define and use any term(s) you wish Joe. You just have to be clear about it. You haven’t done that.

I used a word, I defended my use and now you are desperately trying to hide your ignorance.

You have not been able to defend your use of the word in a specific set theory context. Please do so.

Great. We use the derived relationship to show one is a proper subset of the other. And we should use the same relationship for everything else. To use a contrived relationship is bogus, but you don’t seem to be able to grasp that simple fact.

There are lots and lots of way of setting up a one-to-one correspondence between the positive integers and the postive even integers. It’s the fact that a one-to-one correspondence exists that’s pertinent. As long as we can line up the sets and match the elements one for one then we can decide about things like whether one set is a subset of the other or whether they have the same size.

If we take set Z+ and subtract set E from it, we get another infinite set- the set of all positive odd integers. That alone proves that set Z+ and set E do not have the same number of elements.

Too bad that’s wrong.

Jerad, good at following, not so good at thinking for himself.

If we take set Z+ and subtract set E from it, we get another infinite set- the set of all positive odd integers. That alone proves that set Z+ and set E do not have the same number of elements.

Joe has NOT provided any academic (or otherwise) references to how he wants to use the terms ‘contrived’ and derived’ in a set theory specific context.

I have tried my best to be respond to my objectors and, hopefully, at least as respectfully as they have responded to me.

It is very, very important that y’all respond to this discussion objectively. That is, don’t favour my or Joe’s arguments because of which ‘side’ we’re on. We were discussing mathematical terms. Respond based on what we said not who we are.

And please, check things out for yourself. Google search. Test things out. Don’t just take anyone’s word for it.

As long as we can line up the sets and match the elements one for one then we can decide about things like whether one set is a subset of the other or whether they have the same size.

LoL! There are two different systems used. I am saying we have one and should use it for both. You disagree but cannot provide any reasoning.

Joe has NOT provided any academic (or otherwise) references to how he wants to use the terms ‘contrived’ and derived’ in a set theory specific context.

I used examples along with the common definitions of the words. Jerad doesn’t like that and whines.

A way of modifying a set by removing the elements belonging to another set. Subtraction of sets is indicated by either of the symbols – or \. For example, A minus B can be written either A – B or A \ B.

Hey looks like I was right about that! Jerad disagreed yet sez that I don’t know set theory.

Jerad, The words mean what they do regardless of the setting. That you have some mental block to that fact says quite a bit about you.

In some specific contexts words can take on more specific and narrow meanings.

In English you can refer to the ‘gravity’ of the situation and mean something different from when you use that term in physics.

‘Charge’ in the military means something completely different from ‘charge’ in electricity.

When you use a term implying that is has a specific mathematical usage then you have to be specific about that usage. You have to provide the context. And you have not provided a solid, strict, mathematical definition of ‘contrived’ and ‘derived’ although you implied they had them.

This is not personal. And, if you’ve got the context, it should be easy to provide.

I choose to defend my statements with evidence, logic and reasoning. Only you require the crutch of other people.

Welcome to science. Don’t block the doorways.

So what do you think the match is that is between the same numbers of different sets? Why are you so happy to live inside of your little box?

What if the sets don’t have the same numbers? Consider this:

Is there a one-to-one correspondence between the sets

And only one is natural, meaning it directly matches the same numbers to each other. And yes, that matters.

And no, it doesn’t matter if all you want to do is show that the sets are the same size.

Cuz Jerad sez so? Really??

Nope. I gave you a reference to a whole book about this. You haven’t read it. You refuse to accept that you might be wrong.

Read Kaplansky’s book at least please!! And stop pretending to understand things that you haven’t come to terms with. Things that are commonly accepted parts of mathematics. Ask Dr Dembski if you don’t believe me. He knows. Go on, ask him.

Jerad is on a jihad. Anything that does not come from the materialist Church is automatically wrong.

I, too, am on a jihad. Anything that comes from the materialist Church is suspect. And, as we all know, the materialist Church keeps piling on the crackpottery, nonstop.

When you use a term implying that is has a specific mathematical usage then you have to be specific about that usage.

I have provided the definitions for the words in the context they are being used. I have told you that the meanings of words don’t change just because we are using them to describe what is happening in Set Theory.

Despite my asking multiple times Joe will not or can not provide a suitable, set theory mathematical context for the terms ‘contrived’ and ‘derived’ as they influence evaluating one-to-one correspondences.

I am NOT on a jihad Mapou. I am merely asking that Joe be specific and discipline sensitive to an assertion he made. What’s wrong with that? This isn’t even a major topic. He just claimed that ‘derived’ correspondences were ‘better’ than ‘contrived’ correspondences and I’m trying to figure out what, specifically he means.

Joe, when you get around to actually addressing this terribly simple request I’ll respond. And if you’re not going to bother then don’t waste my or your or anyone else’s time. Time to face the music and dance or leave the floor. It’s up to you.

(I’m pretty sure I know what’s going to happen; Joe is going to buster and bluff a bit more and hope everyone forgets that he dodged a basic question about an assertion that he made. If that’s okay with everyone else then I’ll stop pursuing it because I’ll know that this forum is NOT open and fair and transparent. I’ll know that there’s a double standard. But, I’d really love to be proved wrong.)

Infinitesimals are assumed in Einstein’s physics. It is called continuity. This is the reason for the term ‘spacetime continuum’. General relativity, for example, is based on continuous structures, i.e, infinitely smooth structures. This is a fact. So much so that Einstein had doubts about its correctness as I pointed out earlier:

I consider it quite possible that physics cannot be based on the field concept, i. e., on continuous structures. In that case nothing remains of my entire castle in the air gravitation theory included, [and of] the rest of modern physics. – Einstein in a 1954 letter to Besso, quoted from: Subtle is the Lord, Abraham Pais.

Despite my asking multiple times Joe will not or can not provide a suitable, set theory mathematical context for the terms ‘contrived’ and ‘derived’ as they influence evaluating one-to-one correspondences

And I say that I have done so.

Look I have explained my position such that middle school students understand it. The way I see it the problem is all with Jerad.

Again the example:

Here’s an infinite set, let’s call it Z+ = {1, 2, 3, 4, 5 . . . .}

Here’s another infinite set E = {2, 4, 6, 8, 10 . . . . }

Set E is a proper subset of set Z+ due to a natural match of same numbers. That is it is a derived relationship of a one-to-one correspondence.

To take those same two sets and say that the element placement/ rank can also be a one-to-one relationship (a mapping function is used) is then a contrived relationship.

Infinitesimals are assumed in Einstein’s physics. It is called continuity. This is the reason for the term ‘spacetime continuum’. General relativity, for example, is based on continuous structures, i.e, infinitely smooth structures. This is a fact. So much so that Einstein had doubts about its correctness as I pointed out earlier:

I’m sure you are also aware that Newtonian mechanics does the same thing, that is, it models space and time via smooth manifolds. Is that an issue for you?

Right now I am saying that the set of all positive integers has more elements than the set of all positive even integers.

And that is wrong. It’s been known to be wrong for over 100 years. And yet no one but me points this out.

If you guys want to be stuck with 19th century mathematics it’s okay with me. But I’m disappointed.

This is not controversial. This is not something that people argue about (anymore). There is nothing about this that is part of the ID v evolution debate. There is no reason NOT to call Joe on this.

But no one does.

By the way this:

Set E is a proper subset of set Z+ due to a natural match of same numbers. That is it is a derived relationship of a one-to-one correspondence.

To take those same two sets and say that the element placement/ rank can also be a one-to-one relationship (a mapping function is used) is then a contrived relationship.

is gobbly-gook.

If this is an example of your collective ability to understand mathematics then you’ve already missed the train. How do you expect to even begin to grasp some of Dr Dembski’s points if you don’t get this?

Here’s a question. How can one infinite set be bigger than another infinite set if both have no end? That’s the problem with infinite sets. They only exist in the imagination of mathematicians and materialists.

I agree that infinities do not exist in the universe and I agree that Einstein’s physics is at best very incomplete.

However You and Joe seem to be arguing something much stronger than that. You seem to be saying that infinities are not real. I can’t agree with that.

Physicality and reality are not synonymous

Does that make sense to you?

No, it does not. I’m a yin-yang dualist. I believe that reality consists of two complementary opposite realms, the physical and the spiritual. The former can be created, destroyed or modified. The latter can neither be created, destroyed or modified; it just is. IMO, neither the physical nor the spiritual contains infinite sets of anything. Infinity is an abstract concept. Abstractions do not exist.

Here’s a question. How can one infinite set be bigger than another infinite set if both have no end? That’s the problem with infinite sets. They only exist in the imagination of mathematicians and materialists.

That’s what Cantor established in the late 19th century. His work was extremely controversial at the time but eventually it was shown to be sound and is now part of the structure of mathematics.

Let me try the ‘ID way’ : The creator created Earth as a sphere (if you ignore the ‘bulge’ due to spinning ,which forms a oblate spheroid). Sphere is possible because of a point on infinite plane, so you see, infinity is what allowed creator to create the Earth.

That’s what Cantor established in the late 19th century. His work was extremely controversial at the time but eventually it was shown to be sound and is now part of the structure of mathematics.

I don’t care what Cantor established. It is still hogwash.

I don’t care what Cantor established. It is still hogwash.

Seriously, don’t expect anyone with a modicum of a mathematical background to take you seriously again. This is well established, non-controversial stuff. I learned it at the undergraduate level.

It’s your call but I imagine the folks at The Skeptical Zone will find your position amusing.

Abstract : existing in thought or as an idea but not having a physical or concrete existence.

end quote:

Are you saying that something has to be phyiscal or concrete to exist? You do realize that is the position of the materialist. Is love an abstraction? What about morality?

Joe says

Right now I am saying that the set of all positive integers has more elements than the set of all positive even integers.

F/N: A few balancing words, as we count down the hours to the pre-Christmas Assembly sitting here.

(Forgive the personal note, which is why I have been busy elsewhere. And no, he who abused privilege to slander me will most likely not be present; so the correction on record will be postponed. Cf. here, if you need to know more.)

Now, similar to how x^0 = 1, it is reasonable to accept zero as an even number. Indeed, it has properties that fit with that, and it is in the right place in the number-line to be even: even-odd-even-odd, etc.

(This is probably the easiest way to make it palatable to young students. Besides, it begs us to extend to negative numbers and complex ones. Which reveals my not so hidden agendas here.)

Likewise, because it is useful, we can define transfinite numbers of countable and beyond countable cardinality. Aleph-null is countable, and we can profitably reason:

1, 2, 3 . . .

x 2 . . .

2, 4, 6 . . .

We can go to power sets, and to the issue of the continuum, which is a transfinite number of higher cardinality than the countable one.

These are of course concepts, we have here a basis for defining that the continuum is such that on a number line, for ease of reference, between any two neighbouring values, we may define a third. And, we can note that abstract entities such as 2, 3, 5 and the relationship 2 + 3 = 5 necessarily exist in any world. Based on constructing {} –> 0, {0} –> 1, {0, 1} –> 2 etc, and defining addition and equality appropriately.

From this, we may extend to real world observations as a useful model. We do not need to make any absolute commitment to there being an actual continuum in the physical world, it is fine grained enough to use continuum models in many cases.

Similarly, by defining hyper-real numbers and taking reciprocals, we can give mathematical substance to infinitesimals and justify treating dx, dy, dz, dt etc as tiny all but zero numbers. Non-standard analysis, an alternative foundation to Calculus, lies down that road. (More of my not so hidden agenda . . . and I see Calculus is now in 5th form math in the UK, hint hint hint. For crying out loud, I have long known that it was reported decades back that every High School child in Russia has to do several years exposure to physics and calculus. Can’t we do an introductory Physical Science, Physics and Chemistry sequence and can’t we put some serious math in too? And some logic calculus? Tossing in some computer science too, Hon Mr Santa Claus, sir?)

Of course the usual ZFC set theory anchored, epsilon and neighbourhood, limits oriented approach to calculus can be taught as developing another enriching perspective. (My favourite demonstration is water flowing into a bucket. Add in a leak and we are off to system modelling and controls, thence of course differential equations, Laplace transforms and the complex frequency domain; also the discrete state Z transform equivalent and my favourite heavy rubber sheet with poles and nailed down zeros graphical model. Where, drop off the transient terms and we are in Fourier transform space. Another not so hidden agenda.)

Why am I doing this?

First, to remove the strawman caricature being painted above by those who pretend and prefer that supporters of design theory are scientific and mathematical ignoramuses as a whole.

And, BTW, when I take a bit of time, I want to revisit the Abu 6500 c3 reel as an example of FSCO/I and how it can be informationally quantified using the nodes arcs wiring diagram framework. As in:

I_tot = SUM [I(pi)]

Where, multi-dimensiaonal pseudo-vector pi = {part-ID in space of parts | orientation | location of COM | node ports and connexions}

. . . leading to a virtual configuration space

In short, we here have a descriptive language reducible to bits through a chain of Y/N structured questions. As Orgel stated in 1973, without referring explicitly to bits.

That is, the shoe is on the other foot.

Then, we can look at the ribosome etc as cases in point of FSCO/I.

PS: Remember, how, over the course of several weeks, too many objectors to design theory, were unable to bring themselves to acknowledge that FSCO/I is real, that it was conceptually identified by Wicken and Orgel in the ’70’s as applicable to life, and that it points to a way to quantify the information content of organised functional entities, at least in principle based on structured chains of Y/N questions. Never mind what AutoCAD etc routinely do. As for that such FSCO/I extends to cases in the world of life . . .

First, to remove the strawman caricature being painted above by those who pretend and prefer that supporters of design theory are scientific and mathematical ignoramuses as a whole.

Your flock needs some learnin’ regarding Cantor’s work.

With all due respect, you just used loaded, inapplicable terminology — “flock” — that insinuates mindless following; when in fact we see understandable skeptical concern because of widespread and longstanding major violations of trust in institutions that should be paragons of intellectual virtue.

As in, there is a reason why the term junk science has arisen, and why people reasonably fear Government backed ideologised institutional science.

Let’s start with: E-U-G-E-N-I-C-S.

As in, that’s no phobia.

Or, have you forgotten that Hunter’s Civic Biology, at the heart of the Scopes trial, was riddled with eugenics?

One wonders how comes that inconvenient little fact tends to at best be lost in the footnotes . . .

That, eugenics laws were put on the books, with horrible consequences, and not at all just in Germany?

Indeed, Germany explicitly copied US laws, IIRC esp. those of California?

Do you see now, why for cause people may be inclined to be fairly suspicious?

When trust is abused, it will be forfeited.

As Climate Scientists are currently finding out.

And, as is building up on origins science.

And when trust has been sufficiently abused on a large enough scale, it poisons the atmosphere, leading to questioning anything that seems counter to common sense.

That can become an excess, but the need tyo be critically aware in an age of widespread ideological abuse of science should be frankly faced.

Instead of sneering contempt and loaded insinuations, it would be more reasonable to ground and warrant.

But then, isn’t that what I have been calling for on origins science for two and more years in reply to some remarks you and others made?

With no serious takers to date?

Now, I highlighted that Mathematical models, including continuum hold primarily, conceptual validity. They are extended to real world contexts by use of models and meaning assignments. That this works very well is an astonishment to those whose fundamental view of nature is non rational and arguably irrational.

But, from the days of Newton, Boyle and Kepler, that is no surprise to those who understand our cosmos to have been shaped by an utterly rational architect.

I suggest, lastly, that when one refers to actual numbers, transfinite is a more applicable term, with understanding that such numbers start scaling with aleph-null, the cardinality of the natural numbers and linked sets.

You put your finger on a phenomena that is pervasive in our world today. The complete collapse of the old authority structure

Secularists thought that when people abandoned religious authority it would mean that their own authority figures (scientists and government officials) would replace it.

What is happening instead is anarchy. Each person trusting only those who agree with him.

You see it on both sides of the isle. The materialists here won’t even look at arguments from anyone who is not a fellow traveler for example.

Talk about Déjà vu

quote:

In those days there was no king in Israel. Everyone did what was right in his own eyes.
(Jdg 17:6)

With all due respect, you just used loaded, inapplicable terminology — “flock” — that insinuates mindless following; when in fact we see understandable skeptical concern because of widespread and longstanding major violations of trust in institutions that should be paragons of intellectual virtue.

I thought you’d like the Biblical allusion. Sorry.

Let’s start with: E-U-G-E-N-I-C-S.

Kind of off topic don’t you think?

Do you see now, why for cause people may be inclined to be fairly suspicious?

About stuff they can easily look up for themselves? That’s part of most undergraduate mathematics curriculum?

If they didn’t trust me they could just do a search for themselves or look it up in Wikipedia.

As Climate Scientists are currently finding out.

Waaaaaay off topic now.

And, as is building up on origins science.

Definitely a minority opinion.

And when trust has been sufficiently abused on a large enough scale, it poisons the atmosphere, leading to questioning anything that seems counter to common sense.

You mean like an underfined, undetected designer?

I’m sorry, that was a bit snide. But you get my point? You call me out when I question your beliefs but you defend you compatriots when they question me over something that isn’t even controversial.

I know, you’re going to accuse me of inflaming the situation. But I’ve spent hours trying to explain this stuff which you clearly understand only to be told its hogwash. That such things exist only in the minds of mathematicians and materialists. What?

How about this from Mapou #110

Jerad is on a jihad. Anything that does not come from the materialist Church is automatically wrong.

I, too, am on a jihad. Anything that comes from the materialist Church is suspect. And, as we all know, the materialist Church keeps piling on the crackpottery, nonstop.

Very understanding and respectful eh?

Instead of sneering contempt and loaded insinuations, it would be more reasonable to ground and warrant.

I spent hours being polite, trying to explain, here and on Joe’s blog. And, in the end, I’m told its wrong.

LOOK IT UP!!

Now, I highlighted that Mathematical models, including continuum hold primarily, conceptual validity. They are extended to real world contexts by use of models and meaning assignments. That this works very well is an astonishment to those whose fundamental view of nature is non rational and arguably irrational.

Yes . . .

But, from the days of Newton, Boyle and Kepler, that is no surprise to those who understand our cosmos to have been shaped by an utterly rational architect.

A matter of opinion. I think relativity and quantum mechanics shook things up a bit.

I suggest, lastly, that when one refers to actual numbers, transfinite is a more applicable term, with understanding that such numbers start scaling with aleph-null, the cardinality of the natural numbers and linked sets.

From Wikipedia:

“Transfinite numbers are numbers that are “infinite” in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as “infinite”. However, the term “transfinite” also remains in use.”

So, it was just a term used to ease the ‘pain’ of accepting his work. I don’t know what you mean by they ‘start scaling’. Scaling means making bigger or smaller but it’s pretty clear that now most people accept that transfinite numbers are really infinite.

Jerad, Session now in progress, complete with a lead in by the speaker that tried to justify himself but implicitly conceded my point, looks like the intervention of the Christian Council about angry remarks and attacks was decisive. No time for a full response. I pause to note that transfinite avoids the issue of an actual infinity, which is still a problematic matter, especially for those who imagine they can deliver an infinity in countable succession. As in an infinite, causally successive actual past of inherently contingent beings. KF

PS: I note that the eugenics issue, the Scopes Monkey trial, Hunter’s Civic Biology and linked issues are highly material to the polarised debates that surround origins science, and lurk behind the subtext of contempt that so often crops up when loaded allusions used by objectors. So, if there is an attempt to project the pall of obscurantist fundamentalism, such are immediately, highly relevant. And, I will not shy away from pointing out that fact.

PPS: Try the sequence of patently increased cardinality aleph null, power set thereof, and so forth, with the debate as to what c the continuum cardinality is, often suggested — note the term — to be that of the power set of aleph null.

Oh please, to suggest that all mathematicians agree that some infinities are bigger than others (much less other thinkers besides so called mathematicians), is pure nonsense. Plenty of mathematicians don’t think infinity is a real concept.

If infinity means something going on forever, then playing with words to call one unlimited amount bigger than another unlimited amount, is just playing with words. Its no different than saying “the biggest thing possible”, and then claiming the biggest thing possible plus a little more is even bigger. Cantor doesn’t get to make the illogical logical. Its pure word play bullshit.

If I say 2 x infinity is bigger than 1 x infinity, that doesn’t make it so.

phoodoo: If infinity means something going on forever, then playing with words to call one unlimited amount bigger than another unlimited amount, is just playing with words.

It’s a matter of mathematical proof, not opinion. See Cantor 1874 or Cantor 1891.

Here’s a question. How can one infinite set be bigger than another infinite set if both have no end? That’s the problem with infinite sets. They only exist in the imagination of mathematicians and materialists.

Here’s one way to look at it: Some sets, such as the integers or the rational numbers, can be arranged in an (infinite) list. This diagram shows one scheme for putting the positive rationals in a one-dimensional list.

For other sets, such as the real numbers, this is impossible. Any list of real numbers which you construct will be missing some values. In that sense, the set of real numbers is larger than the set of rationals or integers.

My post #120 was in moderation for a little while; I’d be interested to hear your response.

DaveS (Zacheriel, this is where Cantor went wrong),

What does “missing some values” in your post mean? Missing from what?

That diagram is totally artificial. There are no requirements for what must match up with something from another set.

One set is a bunch of things you can label anyway you want, and another set is also a bunch of things you can label a different way if you choose. There is no universal requirement that one label matches another, and that therefore you can say some labels don’t have a counterpart in the other set.

phoodoo: There is no universal requirement that one label matches another, and that therefore you can say some labels don’t have a counterpart in the other set.

In set theory, a bijection, or one-to-one correspondence, is how we determine if two sets have the same number of elements. We can prove that rationals and the natural numbers have a one-to-one correspondence. We can prove that real numbers and the natural numbers do not have a one-to-one correspondence. In other words, there are more real numbers than can be counted with natural numbers.

DaveS (Zacheriel, this is where Cantor went wrong),

What does “missing some values” in your post mean? Missing from what?

That diagram is totally artificial. There are no requirements for what must match up with something from another set.

One set is a bunch of things you can label anyway you want, and another set is also a bunch of things you can label a different way if you choose. There is no universal requirement that one label matches another, and that therefore you can say some labels don’t have a counterpart in the other set.

By “missing some values” I mean that given a list of real numbers, you can construct another real number which is not on the list. Therefore the list was incomplete.

I don’t know what you mean by “artificial” there. The diagram simply shows one way to arrange all (positive) rational numbers in a one-dimensional list. There are infinitely other ways to arrange them, all of which work just as well.

Try as much as you like, you will never find such a list which includes all the real numbers.

The bijection is totally artificial. I could just as easy color all of the elements in one set red and all of the elements in the other set blue. Then I just match one red with one blue for eternity. Problem solved

Since infinity is a made up concept, which has no tangible reality, it makes just as much sense to ask, If I have peace in one set, and beauty in another, which is bigger?

The reason that the whole argument is an artificially constructed one is because who do you think named the different kinds of numbers? Who decided that some numbers are going to be called whole, and some natural, or real or complex or integers….? Those are completely made up names, you can just as easily call any number you can think of Fred.

So now you have several sets, one is an infinite number of Freds, another set is an infinite number of Freds, plus an infinite number of M&M’s. A third set has an infinite number of jelly beans, plus an infinite number of M&M’s but no Freds. Which set is biggest?

phoodoo: I could just as easy color all of the elements in one set red and all of the elements in the other set blue. Then I just match one red with one blue for eternity.

However, unless you specify the order you place them, you don’t know if you have matched them all up, so you haven’t shown whether or not there is a one-to-one correspondence. However, if we arrange the rationals diagonally, we can show that we have included all the rationals, and that they can all be matched to the natural numbers.

phoodoo: Those are completely made up names, you can just as easily call any number you can think of Fred.

Sure, though we might be more specific and say any number phoodoo can think of is a phoodoo number.

However, we’re not talking about phoodoo numbers, but natural, rational, and real numbers. Each of these types of numbers have unambiguous and explicit mathematical definitions, but you probably intuitively know at least the natural numbers; 1, 2, 3, …

But Zachriel, the definitions are man made, for crying out loud.

Which set is bigger, the set of infinite M&M’s or the set of infinite M&M’s plus infinite jelly beans? Or the set of infinite candy? Jelly beans and M&M’s and candy all have definitions.

phoodoo: Who do you match with whom if I call numbers Fred?

Don’t know Fred. Can he count? Can he do fractions? Decimals?

As for your red and blue matching:

Z: unless you specify the order you place them, you don’t know if you have matched them all up, so you haven’t shown whether or not there is a one-to-one correspondence. However, if we arrange the rationals diagonally, we can show that we have included all the rationals, and that they can all be matched to the natural numbers.

Right now I am saying that the set of all positive integers has more elements than the set of all positive even integers.

And that is wrong.

Prove it.

By the way this:

Set E is a proper subset of set Z+ due to a natural match of same numbers. That is it is a derived relationship of a one-to-one correspondence.

To take those same two sets and say that the element placement/ rank can also be a one-to-one relationship (a mapping function is used) is then a contrived relationship.

is gobbly-gook.

Perhaps to your limited intellect it is. However your limited intellect means nothing to me.

The reason that the whole argument is an artificially constructed one is because who do you think named the different kinds of numbers? Who decided that some numbers are going to be called whole, and some natural, or real or complex or integers….? Those are completely made up names, you can just as easily call any number you can think of Fred.

Well, you can call it artificial if you like, but over the millenia people have found it useful to distinguish between different types of numbers. For example, numbers which can be written as a fraction of integers have nice properties that arbitrary real numbers do not have in general. Integers are even nicer, IMHO.

So now you have several sets, one is an infinite number of Freds, another set is an infinite number of Freds, plus an infinite number of M&M’s. A third set has an infinite number of jelly beans, plus an infinite number of M&M’s but no Freds. Which set is biggest?

Obviously not enough information to go on here! Fortunately a lot (but not everything) is known about the real numbers and certain of its subsets, so comparing the ‘sizes’ of the reals and the rationals is possible.

That’s what Cantor established in the late 19th century. His work was extremely controversial at the time but eventually it was shown to be sound and is now part of the structure of mathematics.

Cantor didn’t establish anything and no one has shown what he sad to be sound. t isn’t even of any use.

The only reason this discussion has any value to me, is simply to show how many so called skeptical thinkers are just willing to accept any old bit of nonsense someone claims, as long as they think that someone is a “scientist” or oooooo, a “mathematician!”

If Cantor says two infinite sets are different, well then, little be it for you to question, it just must be true. You never even bother to think who gave numbers that name in the first place. You can substitute any name for something else.

An infinite number of M& M’s is no different than an infinite number of blue M& M’s or an infinite number of Red M&M’s plus blue M&M’s. Just because I start adding more names to one set than to another, that doesn’t make one set bigger. Clearly if I used Cantors logic, an infinite number of blue and red M&M’s would be more than just an infinite number of blue M&M’s because there is nothing to pair up the red M&M’s with.

It just goes to show you how little people actually think, but instead just accept the word of authority.

“All numbers” is not well-defined. Do you mean the real numbers or the natural numbers or something else?

phoodoo: Infinite M&M’s = Infinity

No, that is not correct, otherwise an infinity of shoes would equal an infinity of M&Ms, which is clearly not the case. Rather, they have the same cardinality. That would be written | infinity of M&Ms | = infinity or | infinity of shoes | = | infinity of M&Ms |.

phoodoo: If Cantor says two infinite sets are different, well then, little be it for you to question, it just must be true.

No, it’s because Cantor *proved* that the real numbers do not have a one-to-one correspondence with the natural numbers.

Right now I am saying that the set of all positive integers has more elements than the set of all positive even integers.

And that is wrong.

Prove it.

I gave one approach in 75, by using ranking. Another way to do it is to divide the positive even integers by 2. You then get back to the positive integers.

Infinity is tricky, and I certainly don’t understand a lot of the subtleties, but I wouldn’t reject areas of mathematics just because I don’t understand them.

I gave one approach in 75, by using ranking. Another way to do it is to divide the positive even integers by 2. You then get back to the positive integers.

Infinity is tricky, and I certainly don’t understand a lot of the subtleties, but I wouldn’t reject areas of mathematics just because I don’t understand them.

What set theory are you talking about? Joe has extended the set theory with his own definitions. So if you’re not talking about “Joe’s set theory”, then your approach is irrelevant.

phoodoo: Which set is bigger, an infinite number of blue M &M’s or an infinite set of blue and red M&M’s?

They are the same cardinality as there is a one-to-one correspondence between the sets.

phoodoo: Cantor proved nothing.

You say that, but haven’t shown it. Generations of mathematicians have found Cantor’s proofs convincing. It will take more than handwaving to discount it.

From my understanding infinity isn’t a number, it is a journey. Now if you are on a train (Einstein’s train) that had two counters, counter A counted every second of the journey and counter B which counted every other second of the journey. Counter A would represent the set of positive integers and Counter B would represent the set of all positive even integers.

At every second throughout the journey Counter A would have a higher count than Counter B, meaning set A would always have more elements than set B.

What my detractors seem to be saying is that counter B should count double every time it counts to allow it to be the same as counter A and because of that they are equal at every other second, and that is the only time you are allowed to look.

Generations of mathematicians have found Cantor’s proofs convincing.

His “proofs” are contrived and not derived. For example the derived relationship is that which is used to determine if one set is a proper subset of another. The contrived relationships disregard the natural matchups and because of that show a false relationship.

If you draw a straight line with all the blue M&Ms on one line, and another line with all the blue M&M’s from the second set and a third line with all the red M&M’s, you can match all the blues from one set with all the blues from the other set, but you still have all the reds left over, so clearly the red and blue set is bigger, right?

phoodoo: If you draw a straight line with all the blue M&Ms on one line, and another line with all the blue M&M’s from the second set and a third line with all the red M&M’s, you can match all the blues from one set with all the blues from the other set, but you still have all the reds left over, so clearly the red and blue set is bigger, right?

There is a one-to-one correspondence. For each M&M in the first set, you map alternating between the second and third sets. To show the sets are different sizes, you have to show there is no such one-to-one correspondence, not merely that you can’t think of one.

Could you provide a link or name the source from which you learned the concepts of contrived and derived relationships in set theory? You have piqued my curiosity. Thanks.

Infinitesimals are assumed in Einstein’s physics. It is called continuity. This is the reason for the term ‘spacetime continuum’. General relativity, for example, is based on continuous structures, i.e, infinitely smooth structures. This is a fact. So much so that Einstein had doubts about its correctness as I pointed out earlier:

I’m sure you are also aware that Newtonian mechanics does the same thing, that is, it models space and time via smooth manifolds. Is that an issue for you?

Certainly. Newton was a big fan of the continuity nonsense but he did not start it. The Greeks started it thousands of years ago with their silly concept of a line having an infinite number of points. Newton was a genius but he did not know everything. I don’t think he even realized that his inverse square law broke down at short distances. But he was right about the universe being absolute. He also understood that there was a force or causal principle responsible for inertial motion. He could not prove it, so he attributed it to God and ignored it in Principia. In this respect, Newton (and even Aristotle before him) was light years ahead of the relativists and everybody else.

I don’t care what Cantor established. It is still hogwash.

Seriously, don’t expect anyone with a modicum of a mathematical background to take you seriously again. This is well established, non-controversial stuff. I learned it at the undergraduate level.

It’s your call but I imagine the folks at The Skeptical Zone will find your position amusing.

LOL. I don’t give a rat’s asteroid. And I am nobody’s female dog. 😀

The idea that two infinite sets can be compared to see which one is bigger than the other is stupid on the face of it. The only way to compare two series is to know their sizes. An infinite sets has no size that anybody can put a finger on. These so called one-to-one comparisons of infinite sets is ludicrous. Only materialist morons will swear up and down that they have anything to do with sizes.

Folks, got a moment to follow up. Let’s start with a basic point, Mathematics is an intellectual, logic puzzle game that did arise from practical considerations but has gone well beyond that. Probing and arguing has led in many fruitful directions, some of which were not conceived of when the original ideas were put down. And, it turns out that mathematical models have great utility in dealing with the real world. For example infinitesimals and calculus, whether we go by standard or non standard approaches. In short, lighten up. By certain accepted rules, it makes sense to speak in terms of the continuum being of higher cardinality than the natural numbers, it being not possible to put them into one to one correspondence. Yes, strange oddities do come out, but they are accepted on the implied logic. And let us remember, at no one point can we show our axioms to be mutually coherent, much less can we set up sets of axioms that are coherent that also point to all true claims in the field, by whatever notion of truth is relevant at this point, perhaps that on other axioms, they would follow. Bottomline, lighten up on Math. And, while many of us are forced to accept Math claims on authority, there is a context of a discipline that is a lot more rigorous than say, origins narratives. KF

PS: That God is not limited other than by the sort of things that are incoherent [God cannot make a square circle out of a paper clip . . . ], or inconsistent with his maximally great being character, has nothing to do inherently with the issues of transfinite numbers and infinitesimals, what continuity is or is not, etc.

I didn’t want to give the impression I was abandoning the conversation but for the next few days I only have an iPad and, lovely as they are, without a keyboard I find copying and pasting large quantities of text painful in the extreme. I’ve pretty much said my piece.

I’m sure the arithmetic of ‘infinities’ sounds very bizarre at first but it IS well developed and non-controversial now. And it is not just a few pointed-headed geeks with no tans or lives saying it. I find it much easier to grasp than quantum mechanics or relativity

Any set which can be lined up with the set of the positive integers so that each element of each set is uniquely matched up with an element of the other set (a one-to-one correspondence) is said to be countably infinite. So the positive even integers is the same size as the positive integers as is the set of all the primes and the set of the rational numbers and the multiples of three, etc. And, yes, if you take the positive integers and take away the evens you still have a countably infinite set of odds left. If you take the positive integers and take away the multiples of 4 you still have a countably infinite set. This may feel contrived or mere assertion but it’s not. AND you can look it all up online!! It’s not hidden or part of an agenda. Dr Dembski will be very familiar with this as will Dr Sewell.

Anyway I won’t be around much over the next few days. I hope you all have a nice Christmas!! I apologise for any typos or garbled text.

Joe @ 162 – I didn’t see any refutations, just something about things being contrived not derived. But I can’t see what the mathematical definition of “contrived” is, so I don’t know what your proof is.

I’m curious – are you saying that if I take every positive even number, and divide it by 2 I don’t get the integers? Can you tell me which positive even number this isn’t true for?

phoodoo: There is only two sets. One which contains only blue, and one which contains blue and red.

Yes, we understand, but you divided the second set into two sets, each of which are presumably infinite.

phoodoo: There is a one to one correspondence of blue M&Ms to blue M&M’s.

Your assignment left out elements, but you might be able to append them to the end after the ellipses. There’s no way to know based on your assignment. We showed you how to form a one-to-one correspondence so that there is no ambiguity.

Mapou: The idea that two infinite sets can be compared to see which one is bigger than the other is stupid on the face of it.

Cantor showed that the set of real numbers can’t be mapped in a one-to-one correspondence with the natural numbers. They are a bigger set.

Mapou: These so called one-to-one comparisons of infinite sets is ludicrous.

That’s how we compare the size of sets, is through a one-to-one correspondence.

Mapou: The idea that two infinite sets can be compared to see which one is bigger than the other is stupid on the face of it.

Cantor showed that the set of real numbers can’t be mapped in a one-to-one correspondence with the natural numbers. They are a bigger set.

Nonsense. Correspondence has nothing to do with the size of an infinite set since there is no size. You people can delude yourself till kingdom come but the truth is the truth.

Mapou: These so called one-to-one comparisons of infinite sets is ludicrous.

That’s how we compare the size of sets, is through a one-to-one correspondence.

I don’t care how you compare the size of sets. It’s BS on the face of it. In my computer programs, I simply compare their lengths. This is the way it’s done by logically minded people. Wake up, Zacky-O.

Mapou: Of course it doesn’t and that is the point.

So because your finite machine can’t count to infinity, infinity doesn’t exist? Then why did you use the term “ad infinitum” above?

“ad infinitum” is a term that means that something can grow or extend indefinitely. It does mean that infinity exists. All sets are finite, period. Wake up Zacky-O.

Natural numbers can grow indefinitely but that does not mean that you can have a set of natural numbers. If you can’t create a set on any conceivable computer, it does not exist.

Mapou: If you can’t create a set on any conceivable computer, it does not exist.

So now it whether the computer is conceivable. We can conceive of an infinite computer, so that’s not a problem.

ETA: For instance, Turing conceived of “..an unlimited memory capacity obtained in the form of an infinite tape marked out into squares, on each of which a symbol could be printed.”

Zacky-O, the idea that either you or Turing can conceive of an infinite computer is 100% bogus. Your brain is very finite and, in your case, very limited.

Try conceiving of an infinite number of points on a line. Count every single one of them and report back to me when you’re done. I’ll wait.

I’m on a tablet, so my apologies for not cutting and pasting. Regarding two of your posts:

1) What do you propose be done about the use of continuous structures in physics? Do you have an alternative to slot in its place? I think most physicists and engineers understand that these are just models; a straight train track is not literally an interval in the real numbers, for example. And people are able to land spacecraft on comets, after all.

2) Regarding the infinite set issue, you can represent both the integers and the rational numbers on a computer as lazy lists. You cannot represent the real numbers in this way, however, so that’s one way to distinguish between countable and uncountable sets in a computer context.

I just proved you wrong, and it was simple as could be. There is no ambiguity between matching a blue with blue.

Since there is nothing to match the reds up with, the set with reds must contain more right? Its simple, its obvious, and its completely stupid to suggest that just because one can arbitrarily decide to name something on one set, and thus only the ones with the names you prefer can be matched with the other set, that this means one set is bigger.

The whole concept of what you pair with what is an artificial construct, that anyone can do with any set, so its meaningless. But it does have meaning in showing that scientific materialists are as gullible as they come, just because they believe someone with a title tells them something.

But it does have meaning in showing that scientific materialists are as gullible as they come, just because they believe someone with a title tells them something

If you understand the concept of line and line segment, you should understand infinity. Line has infinite length, line segment is part of the infinite line.

I’m on a tablet, so my apologies for not cutting and pasting. Regarding two of your posts:

1) What do you propose be done about the use of continuous structures in physics? Do you have an alternative to slot in its place? I think most physicists and engineers understand that these are just models; a straight train track is not literally an interval in the real numbers, for example. And people are able to land spacecraft on comets, after all.

Actually, nobody in physics uses continuous structures even if they think they do. Landing on comets is not proof of infinity. Calculus is as discrete as can be, a million mathematicians and physicists jumping up and down and screaming otherwise notwithstanding. Anything that uses numbers is discrete because numbers are discrete by definition. The problem is in the refusal to admit that the universe is necessarily discrete. It’s a disastrous mindset because it prevents us from seeing nature as it is. Once you accept that the universe is discrete, you immediately realize that distance (space) is abstract, a perceptual illusion: it does not exist. And it’s not just space. The concept of a time dimension in which we are moving in one direction or another is also a conceptual disaster. It is what prevents us from understanding why nature is probabilistic. I mean, if there is no time, it is impossible for nature to calculate the exact timing of interactions. Nature is forced to use probability in order to obey conservation laws in the long run. This is the reason that particle decay is probabilistic. It has nothing to do with such silly notions as state superposition.

2) Regarding the infinite set issue, you can represent both the integers and the rational numbers on a computer as lazy lists. You cannot represent the real numbers in this way, however, so that’s one way to distinguish between countable and uncountable sets in a computer context.

Not true. Nobody, I repeat, NOBODY, can model the infinite. It is a form of nerdish delusion, pure unmitigated hogwash. Why? Because whatever model you use is finite.

I understand the concept of a line just fine me_thinks.

Now do you understand, the concept of a human creating what matches in one set with what matches in another set? It is a convenience. It is a way to organize ones own understanding. But it has nothing to do with the actual elements in that set.

One doesn’t have to use Cantors preferred way of choosing what matches with what. One can match the number 5 in one set, with the symbol.333333333 if that is what one chooses. Next they match -6.2 with 1/2.

There is no reason to prefer one matching system with another, other than it might make it easier for you to remember. As long as both sets contain an infinite number, you can always match one with another for infinity.

The sets contain an equal amount.

Some concepts in math are simply created to make it easier to explain a concept, that doesn’t make them reality or fact.

Is a point in geometry infinitely small? If it is, it is also infinitely large, therefore it has no size. You can not measure something with no size, so we must create size in our brains, to help to understand reality. Convenience and reality are two different things. Don’t let others always do the thinking for you.

Calculus is discrete? You have a very broad definition of discrete. But that makes me wonder what your issue with physicists using smooth manifolds is, if they are discrete objects. Sounds like the perfect structure for modeling a discrete universe!

Calculus is discrete? You have a very broad definition of discrete. But that makes me wonder what your issue with physicists using smooth manifolds is, if they are discrete objects. Sounds like the perfect structure for modeling a discrete universe!

Physicists do not use smooth manifolds even if they think they do. They use a discrete and finite computer to do their calculations. It would take an eternity to calculate smooth manifolds for the simple reason that infinite smoothness doesn’t exist and cannot be computed. As soon as one gets close to the discrete Planck scale, all calculations that assume smooth manifolds begin to break down. Take the inverse square law of gravity, for example. It breaks down when r approaches 0. If it didn’t, all particles would collapse under the force of their own gravity.

Let’s suppose an electrical circuit is associated with the equation

[3sin(10 – t)]/(10 – t)

And you want to know what’s going on at t = 10. If you plug in t = 10 you get 0/0 ( which is called an indeterminate form) and that makes no sense to you.

So you graph the function and it looks ‘okay’ around t = 10. Cantor’s work gives us a rigorous grounding for dealing with such situations mathematically. In other words: NO HAND WAVING. We get solid, dependable analytic techniques.

(In this case 0/0 does not come out to be 1 although that some times does happen. Graph the function and see!! Be careful with the brackets/parentheses one mistake and it all turn out wrong

Three times the sine of the quantity10 – t and all of that divided by 10 – t

Mapou: Once we remain in the province of analysis and algebra on variables, calculus is continuous, though you are quite correct that on moving to the world of finite computation, things have gone discrete. And the parallels between difference and differential equations should give pause. However, there is in fact an analytical bridge between Laplace and Z transforms that brings in dynamical elements when physically instantiated; where of course the famed imaginary quantity, sqrt(-1), is material to the analysis, which becomes a way to do 2-d vectors algebraically . . . bringing in forcefully, the point that the math analysis applies to an ideal world of “forms” that gives results we find useful to extend to our everyday real world, and raising issue of the the power of conscious contemplative insightful meaningfully reasoning mind vs blind mechanical GIGO-driven computation. The old fashioned way of saying much of that is to say we do digital stuff using analogue components. The effect of a power supply glitch on a supposedly discrete state system is a classic example. (And I still sometimes reflect on the short-cut that worked, using a silvered mica cap instead of the usual ceramic for power supply decoupling for such a case, and yes the former worked when the latter failed.) KF

PS: I still swear by RPN logic HP calculators, all the way from a fondly remembered HP 21 to my current HP 50, with a couple of HP 48 emulators snuck in there. And, I have a very good report on a Victory, Hecho en la Chine knockoff on the HP 12, for my other set of hats. IIRC the Russians long used their own RPN logic calculator family (Elektronika MK-61 and kin) as last ditch backup for spacecraft navigation.

PPS: Design objectors, you need to recall that Mapou comes down pretty much on your side of the divide on brain-mind issues, if you pardon a few approximations.

PPPS: Anyone who imagines that design thinkers are in lock-step indoctrination should now be seriously re-thinking such crude caricatures.

Jerad, nope, it’s L’Hopital’s Rule. And while Cantor gives one way to look at it, there’s more than one way to skin that cat-fish analytically. And “hand-waving” is dismissive rhetoric. I’d say, that we have various approaches that answer reasonably to the matter, and that it would be helpful to take a look at the implications of Godel’s findings, for schemes that imagine Mathematical, axiomatic systems as a whole to have anything like utter certainty. KF

PS: And, do I need to point to self-evident truths . . . which seem to give vapours to many objectors to design thought?

Mapou, you have hit on one of the interesting points in physics, where a point particle shows itself an ideal, and gives warning on limits. I’d actually say, before we get to infinite self-collapse, we get to everything being a tiny black hole as escape velocity hits c. Where the space-time fabric rips and we are in a different world than we think. Then think about say an electron with its charge, and ask why it does not fly apart under mutual repulsion, as we shrink the presumed ball ever smaller. Oh yes, that does point to the issue that pushing to the all but zero is a mathematical, idealised exercise . . . which has been my point. Classical results here point beyond themselves. Presto, the need for something shaped pretty much like quantum theory and Relativity, with all sorts of interesting paradoxes and unresolved issues. If Godel has mathematicians walking by faith in what they do not see, wait till the physicists come in the door. As in, we all walk by faith and not by sight, the question is in what, why. KF

PS: How do you tell the physicists? Wait till it gets dark, we are the ones that glow faintly blue-green.

But Cantor put things like L’Hospital’s Rulr on a firmer foundation.

My use of the term ‘hand waving’ was in response to a previous commentor who used it first!! The accusation started on the other side. I’m just trying to help by giving an example that hadn’t been mentioned.

Jerad, define firmer in a post Godel world, with paradoxes of the infinite lining up to come in the front door. And, I am not so sure about firmer footings when we have now got a context for seeing why infinitesimals worked so well to get Calculus going. But, of course, I see no reason to dismiss Cantor’s work on the transfinite. Where, I insist, transfinite is a useful term that does not carry the baggage that infinite does. Indeed, it allows us to give some substance. But every time I see those triple dots, I see how we say, we extend to infinity with the eye of faith as we cannot actually deliver the infinite in steps. We are only, ever laying out a logical, in-principle case. Somewhere out there the ghost of Plato is laughing uncontrollably and gasping out that he told us so about forms long since. KF

PS: Think about the challenge that non-being, a genuine nothing can have no causal powers; so if there ever was nothing — the ultimate zero — there ever would be utterly nothing, but here we are. Then, think about a suggested infinite succession to the present of causal steps and entities. Then, contrast a contingent cosmos joined to an underlying necessary being. And, ponder the even deeper mystery, eternity. The logic involved is pointing to some pretty deep waters.

I’ll just stick with the math I think. I don’t grok an ultimate zero. And I’m not a Platonist, i don’t believe in ideal forms or states.

Godel said we can’t discover everything that’s true about a system by merely piling up theorems on top of axioms. Sometime you have to step out side the system and work back down. But, by then, Cantor’s work was firmly esconsed in the structure of modern mathematics. There is no reason to re-evaluate that which has been previously discovered and built upon. Math isn’t like physics, the old stuff still works just as well as it ever did. That’s why math has theorems instead of theories. Maybe theorems are as close as we can get to ideals eh?

When you say transfinite I’ll just think infinite.

Jerad: I take it you are aware that we are here dealing with an ideal conceptual world that we view as holding a reality that is powerful enough to shape and influence how we analyse and understand the world. Plato’s ghost is laughing, laughing uncontrollably as he looks on at materialists grasping with a world of mathematical and logical realities that point to overturning their preferred vision of reality. Next, Godel’s work had two sides in this context, that not only for rich domains, are there truths unreachable from any given set of coherent axioms but that there is no constructive process to create a set of even admittedly limited axioms that are known ahead of time to be coherent. So, we must recognise the now almost 100 years long state of play with mathematics, which yes has theorems but in a revolutionised context post Godel. And, I do not reject Cantor’s work, I just think we need to appreciate the difference between an ideal world of concepts and the physical one, and how extensions will break down. I remember pondering on the gravity of a point mass in 6th form physics, and also point masses with pendulums. That’s a gateway to seeing the point. Mapou has a point, though of course I have some most profound differences of opinion and views with him. KF

I see Godel pointing out something about how the wall has been and can be built. I’m not a research mathematician so I don’t see his work affecting my use of theorems and techniques already established or, in the case of theorems, proven.

We do see some aspects of reality much differently. Which is fine. I just can’t see down your path and mine is sad and limited to you. But there’s enough overlap in our ‘worlds’ that we can agree on much actually. And that’s good. In some ways you do have more hope than me but I find my world view satisfying, exciting and to be savoured. I too see a beautiful world despite all the sham, pain, waste and broken dreams. It’s all part of the whole. And I’m trying to see the whole from a different perspective than you. But we are, kind of, trying to head to similar destinations. We don’t have to walk in each other’s footsteps to appreciate the other person’s journey.

Gödel’s Incompleteness Theorems
First published Mon Nov 11, 2013

Gödel’s two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent). These results have had a great impact on the philosophy of mathematics and logic.

In short, we have a situation where mathematical proofs have a massive, pivotal proviso, that those of us who use Math should be aware of.

Yes, even Mathematicians walk by faith and not by signt, in the end.

I didn’t see any refutations, just something about things being contrived not derived. But I can’t see what the mathematical definition of “contrived” is, so I don’t know what your proof is.

What a joke. As if those words require special mathematical meaning.

I’m curious – are you saying that if I take every positive even number, and divide it by 2 I don’t get the integers?

Look, if set A has all of the elements of set B AND it has elements that set B doesn’t have, then set A has to have more elements than set B. That is if we can subtract set B from set A and get another set, then set A has to have more elements than set B.

Please do tell as I am unaware that I am doing such a thing. So I would say that you are making that up.

Joe@169: The contrived relationships disregard the natural matchups and because of that show a false relationship.

How is that changing the rules? Are you saying that we do not use the relationship I posted to determine if one set is a proper subset of another? Do you not understand English?

From my understanding infinity isn’t a number, it is a journey. Now if you are on a train (Einstein’s train) that had two counters, counter A counted every second of the journey and counter B which counted every other second of the journey. Counter A would represent the set of positive integers and Counter B would represent the set of all positive even integers.

At every second throughout the journey Counter A would have a higher count than Counter B, meaning set A would always have more elements than set B.

What my detractors seem to be saying is that counter B should count double every time it counts to allow it to be the same as counter A and because of that they are equal at every other second, and that is the only time you are allowed to look.

If my detractors cannot handle that then it is obvious they don’t know jack about infinity.

we extend to infinity with the eye of faith as we cannot actually deliver the infinite in steps. We are only, ever laying out a logical, in-principle case. Somewhere out there the ghost of Plato is laughing uncontrollably and gasping out that he told us so about forms long since.

I say.

Amen and exactly!!!

Jerad says,

In some ways you do have more hope than me but I find my world view satisfying, exciting and to be savoured.

I say,

Your world view is only satisfying if you ignore all those pesky edges. The problem is all the interesting stuff is found at the edges and the edges have a way of intruding on you when you least expect it like when you are looking at animated line graphs 😉

Folks chained in the cave do a pretty good job of ignoring what is going on outside most of the time but outside is where all the action is. Plus there are all those nagging questions about where those cool dancing shadows come from in the first place.

Remember when I told you you would not follow the evidence if it meant giving up what you hold dear. This is exactly what I meant.

Do the physicists need to do anything differently, then? If they are already using discrete methods, is there a problem?

Now I understand that a quantum theory of gravity is not here yet, so that presumably is a problem you think needs solving.

But if your ideas were integrated into physics, what specific changes would need to be made? If, as you say, time and distance are illusions, what fundamental quantities would remain? We still make use of clocks and measuring tapes, so they must measure something, right?

Maybe a specific example would be helpful. Suppose you drop a 1 kg stone off a 10 meter building. If time and distance are illusory, what actually happens in your view? Can you describe it mathematically? Would you still use F = ma?

Mapou: Physicists do not use smooth manifolds even if they think they do. They use a discrete and finite computer to do their calculations.

You are confusing physics and engineering. Physicists have been using smooth manifolds since Newton and before. While computers have become increasingly important, solving equations, including those that involve a continuum, was how much of physics was done in the past, and is still done. Indeed, even a Euclidean line is infinite in extent, and a Euclidean line segment is infinitely divisible.

Folks chained in the cave do a pretty good job of ignoring what is going on outside most of the time but outside is where all the action is. Plus there are all those nagging questions about where those cool dancing shadows come from in the first place.

Philosophy is very good subject for killing time – make up something and talk about it as if it is profound. The philosophy of philosophy is to make sure that whatever is made out to be philosophical should be of no practical use.

Z, once we move beyond analysis and algebra and correlate variables to the real world, we do have to engage finite round-offs, error propagation in such, statistical sampling issues etc. Mapou is right to highlight that in our calcs and measurements when we touch down in the real world, we do go to things that are inherently discrete state and non-continuous. Those triple dot ellipses again. Just, we try to manage our round-offs very carefully to avoid things popping up that bite us really badly. For Newton, his original value on earth-moon forces was about 10% as rounded off and in light of available measures. KF

PS: Was it the old trick to take a calculator and feed in 0.5 then take sqrt or the like over and over again say ten times, then do the inverse and see if you get back to 0.5?

MT if you think phil is irrelevant to the real world, you are dismissing logic, assessment of knowledge, what there is to be known, and more. Yes, hard questions have no easy answers, but we had better ask them and appreciate our limitations, if we are not to find ourselves int eh snares of ideologies that cannot stand the clear light of day outside the cave. I suggest to you, that Plato’s parable on false enlightenment as an embedded consciousness has much to teach us all. KF

kairosfocus: Mapou is right to highlight that in our calcs and measurements when we touch down in the real world, we do go to things that are inherently discrete state and non-continuous.

That’s not his claim. Mathematicians and scientists work with the continuum and infinity all the time. Models are never complete or perfectly accurate, but the mathematics of the continuum and infinity have been very powerful tools in mathematics and science.

kairosfocus: For Newton, his original value on earth-moon forces was about 10% as rounded off and in light of available measures.

When you calculate the trajectory of a falling object, per daveS’s comment, we reasonably assume the force acts continuously on the object, and that the object moves continuously through space. We can then solve the differential using mathematics based on the continuum. To claim that Euclid, Newton, and Einstein were engaging in “crackpottery”, as Mapou does, is not supportable.

MT if you think phil is irrelevant to the real world, you are dismissing logic, assessment of knowledge, what there is to be known, and more.

Logic,assessment of knowledge is not exclusive domain of philosophy. Maths and science are more logical and require applying logic in practical problem solving, so Philosophy is good mostly for shooting the breeze.

Z: I repeat, M has a point, once we move to the real world of observation, measurement and comparison with expected outcomes, the discrete state comes right in the door. It may not be all that he meant to say — and I have explicitly not endorsed all he says, and indeed have strong differences on ever so many subjects — but it is a point that we all need to reckon with. KF

PS: Whoever pointed out above how an inverse square law would act as the radial distance shrinks towards zero also has made a point we should also attend to.

fifthmonarchyman: How about describing concepts like infinity and the continuum that are “fundamental to Euclidean geometry and to Newtonian mechanics”.

Philosophers grappled with infinity, but made little progress. Suppose you could say philosophers held the problem at bay, but it was mathematicians, such as Georg Cantor, that resolved the fundamental issues.

How about describing concepts like infinity and the continuum that are “fundamental to Euclidean geometry and to Newtonian mechanics”.
will that work for you?

You really think philosopher’s infinity is equal to scientists infinity? Philosophy is not about finding solutions to problems. Scientist don’t consult philosophers before finding solutions.
Most scientists of bygone era indulged in philosophy too, may be that made you to think philosophy is of practical use.

Z, your view of phil seems to be rather narrow. It is a foundational discipline in its own right and has much to say to other disciplines, including both science and mathematics. Indeed the Godel results as above are regarded as phil results, not just mathematical. And, I am by no means so convinced that Mathematicians have fully uncovered the nature of the infinite or more properly the transfinite. Contributions — indeed important ones — yes, covering the whole story, not at all. KF

DS: It is being side-stepped as opposed to disputed. And, the case of the point charge or mass will bear much serious reflection. As, once did the seemingly idle question of taking a ride on a beam of light. Or, comparing a falling apple to the Moon swinging by in orbit. KF

fifthmonarchyman: Since we agree that infinity does not exist in the physical universe how is it that Cantor was not doing philosophy?

Two doesn’t exist in the “physical universe” either. (Edited for clarity.)

If you envelope mathematics within philosophy, then it’s philosophy. If you envelope science within philosophy, then it’s philosophy. If you envelope pinochle within philosophy, then it’s philosophy. You may want to define philosophy.

We’re not against philosophy, by the way. A couple of millennia ago or so, philosophy was quite refreshing.

fifthmonarchyman: How exactly is it different.

Among other things, there’s more than one cardinality of infinity.

In philosophy, infinity can be attributed to infinite dimensions, as for instance in Kant’s first antinomy. In both theology and philosophy, infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno’s paradoxes. In Greek philosophy, for example in Anaximander, ‘the Boundless’ is the origin of all that is. He took the beginning or first principle to be an endless, unlimited primordial mass (???????, apeiron). In Judeo-Christian theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In ethics infinity plays an important role designating that which cannot be defined or reduced to knowledge or power.

Do you see any thing in there which leads you to ‘divergence’ ? (which is mathematical calculated infinity)

I believe it is people like you and Me-think that want to segregate and exclude philosophy from life because it supposedly does not deal in the testable ie phyiscal.

For me knowledge is knowledge no matter how it is acquired.

me_think says,

Do you see any thing in there which leads you to ‘divergence’ ?

I say

No, infinity is infinity no mater if we are talking about God or the total number of primes

fifthmonarchyman: Maybe you might want to define philosophy.

We use the traditional definition, “the study of ideas about knowledge, truth, the nature and meaning of life, etc.”

fifthmonarchyman: I believe it is people like you and Me-think that want to segregate and exclude philosophy from life because it supposedly does not deal in the testable ie phyiscal.

Not sure why you would say that. For instance, ethics is an area of philosophy, and is intimately connected to the human condition.

fifthmonarchyman: No, infinity is infinity no mater …

Well, it turns out that not all infinities are the same cardinality. Some infinities are bigger than others, much much bigger.

fifthmonarchyman: Mathematics when it deals in infinities fits in there rather nicely as well. Always has.

What is the distinction between infinity-ness and two-ness in terms of philosophy? What does philosophy have to say about infinity-ness or two-ness that is not subsumed within a mathematical framework?

Do the physicists need to do anything differently, then? If they are already using discrete methods, is there a problem?

Now I understand that a quantum theory of gravity is not here yet, so that presumably is a problem you think needs solving.

But if your ideas were integrated into physics, what specific changes would need to be made? If, as you say, time and distance are illusions, what fundamental quantities would remain? We still make use of clocks and measuring tapes, so they must measure something, right?

Measurement does not change. It is automatically constrained by the discreteness of reality. We cannot go past the discrete Planck scale. What changes is our understanding of the universe. As I said earlier, the idea of a continuous universe is a disastrous mindset because it prevents us from seeing nature as it is. Once you accept that the universe is discrete, it opens up a whole new vista on reality, a view that was heretofore hidden to us. You immediately realize that distance (space) is abstract, a perceptual illusion: it does not exist. Only particles and their properties exist. The idea that we are moving in space (a la Newton) or spacetime (a la Einstein) is 100% hogwash in the not even wrong category.

Once you realize that there is no distance, then it is obvious that nonlocality is the rule, not the exception. Einstein’s “spooky action at a distance” objection to QM can be immediately dismissed as the product of ignorance.

We can expand on this further. If distance/space is an illusion, then position is not a property of space but a property of the particles themselves. And if position is an intrinsic property of particles, then they are obviously absolute and the idea that position and motion are relative is just bogus nonsense, pure crackpottery of the worst kind. And if position is an intrinsic variable property of particles, there is no reason that they must only change from one value to an adjacent value. In the not too distant future, we will have technologies that will allow us to instantly move from anywhere to anywhere.

It goes much further. If distance does not exist, motion consists of just changing the positional properties of particles. Every change is an effect that requires a cause. The idea that inertial motion requires no cause is thus stupid and dumb. It follows that we are moving in an immense lattice of energetic particles without which there could be no motion. One day, we will learn how to tap into the lattice for energy production and propulsion. We will have unlimited energy production and vehicles that will travel at tremendous speeds without any visible means of propulsion. I foresee an amazing time of free unlimited energy, floating sky cities impervious to weather, earthquakes, tsunamis and other natural disasters, New York to Beijing in minutes, Earth to Mars in hours, billions of robots zipping around the earth and the solar system, etc. This is our future.

I’ve written about this stuff on my blog. Here are a few links:

PS. I will not go into gravity because it assumes that we exist in a 4-dimensional lattice and that we are moving in the fourth dimension at c. Gravity is caused by our interactions with the lattice: it is nature correcting violations to the conservation of energy. It gets a little complicated and this forum is not the place for it.

fifthmonarchyman: Both 2 and infinity are abstract platonic forms they just represent different quantities.

A couple of millennia ago or so, philosophy was quite refreshing.

fifthmonarchyman: It seems to me that all mathematics at least that beyond simple concrete arithmetic is part of the “philosophical framework”?

If you subsume mathematics within philosophy, then it’s philosophy. If you subsume science within philosophy, then it’s philosophy. If you subsume pinochle within philosophy, then it’s philosophy. You may want to define what you mean by philosophy.

You may want to define what you mean by philosophy.

I say,

“the study of ideas about knowledge, truth, the nature and meaning of life, etc.”

Philosophy is simply the love of wisdom.

At some point materialists like yourself have decided that you don’t care about “wisdom” but are only concerned about the purely practical and any thinking beyond that was seen to be old fashioned fruitless nonsense.

The problem is as we see with Cantor and infinity, philosophical wisdom keeps intruding on the practical and everyday.

In the cave it’s the edges are where all the action is and the edges keep showing up in the darnedest places

Merry Christmas

quote:

Wisdom cries aloud in the street, in the markets she raises her voice; at the head of the noisy streets she cries out; at the entrance of the city gates she speaks: “How long, O simple ones, will you love being simple? How long will scoffers delight in their scoffing and fools hate knowledge?
(Pro 1:20-22)

fifthmonarchyman: At some point materialists like yourself …

We’re not a materialist.

We asked you how philosophy informs us about infinity-ness and two-ness, and you referred to a philosopher from 2400 years ago. Our response was and is, a couple of millennia ago or so, philosophy was quite refreshing.

How is that changing the rules? Are you saying that we do not use the relationship I posted to determine if one set is a proper subset of another? Do you not understand English?

A “contrived relationship” is irrelevant for set theory. You’ve extended the set theory with the “contrived relationship” specification.

Do you not understand English?

I’m an engineer. It’s not about understanding English, but about using a domain-specific glossary.

Why do people think that Cantor is an infallible god?

I don’t know of anyone who does.

What good or what use is it to say that all countable and infinite sets have the same cardinality?

It’s an important and counterintuitive advance over the idea that they have different cardinalities.

What would happen if my claims are correct and Cantor is wrong? (besides math books being changed)

Your claims lead to inconsistencies. Mathematicians prefer consistency to inconsistency, for obvious reasons. ‘Joe Math’ is inconsistent and therefore stillborn.

From my understanding infinity isn’t a number, it is a journey.

If you insist on using that metaphor, then infinity is the entire journey. Your mistake is that you keep referring to finite portions of the infinite journey as if they were infinite.

Now if you are on a train (Einstein’s train) that had two counters, counter A counted every second of the journey and counter B which counted every other second of the journey. Counter A would represent the set of positive integers and Counter B would represent the set of all positive even integers.

I was wondering when ‘choo-choo math’ would make its appearance. But if you’re simply counting elapsed time in seconds, what’s the point of the moving train?

Here’s a crucial point, Joe, so please pay attention: At any finite point in time, counter A would represent the finite number of seconds that had elapsed, while counter B would represent the finite number of two-second intervals that had elapsed. Finite. Finite. Finite. Not infinite.

At every second throughout the journey Counter A would have a higher count than Counter B,

After the first second, yes.

meaning set A would always have more elements than set B.

No. You are thinking of sets A and B as finite sets that are continually growing.

They are not finite. They are infinite. They are not growing toward infinity. They already are infinite.

What my detractors seem to be saying is that counter B should count double every time it counts to allow it to be the same as counter A and because of that they are equal at every other second, and that is the only time you are allowed to look.

No, what your detractors are saying is that infinity is not a number. After T=1, there will never be another point in time when counter A is equal to counter B.

Your mistake is to think that counter A and counter B are both counting up to some humongous number called ‘infinity’. They aren’t. Infinitity is not a number.

When we say that your two sets, the positive integers and the positive even integers, have the same cardinality, we are saying that they can be placed into a one-to-one correspondence.

We are not saying that your two counters will converge on the same number.

What good or what use is it to say that all countable and infinite sets have the same cardinality?

It’s an important and counterintuitive advance over the idea that they have different cardinalities.

What’s the importance? If we listen to Occam counterintuitive is most likely wrong.

What would happen if my claims are correct and Cantor is wrong? (besides math books being changed)

Your claims lead to inconsistencies.

That you manufacture. However t is inconsistent to use one methodology to determine if one set is a proper subset of another and then contrive a methodology to see if they have the same cardinality.

It is inconsistent to say that one set as all of the elements of another PLUS elements the other doesn’t have and then say they have the same number of elements.

Not that keith will understand that rebuttal of his “answers”.

a couple of millennia ago or so, philosophy was quite refreshing.

I say

check it out

quote:

Barfield never made me an Anthroposophist, but his counterattacks destroyed forever two elements in my own thought. In the first place he made short work of what I have called my “chronological snobbery,” the uncritical acceptance of the intellectual climate common to our own age and the assumption that whatever has gone out of date is on that account discredited.

You must find why it went out of date. Was it ever refuted (and if so by whom, where, and how conclusively) or did it merely die away as fashions do? If the latter, this tells us nothing about its truth or falsehood. From seeing this, one passes to the realization that our own age is also “a period,” and certainly has, like all periods, its own characteristic illusions.

From my understanding infinity isn’t a number, it is a journey.

If you insist on using that metaphor, then infinity is the entire journey.

Why is it a metaphor? And the journey never ends so there isn’t any “entire journey”.

Your mistake is that you keep referring to finite portions of the infinite journey as if they were infinite.

Your mistake is thinking that is what I do. But then again you love to hump a strawman.

But if you’re simply counting elapsed time in seconds, what’s the point of the moving train?

So children like you could play too.

At any finite point in time, counter A would represent the finite number of seconds that had elapsed, while counter B would represent the finite number of two-second intervals that had elapsed. Finite. Finite. Finite. Not infinite.

At EVERY finite point in time, forever. Forever. Forever. Forever- infinity even.

You are thinking of sets A and B as finite sets that are continually growing.

They are not finite. They are infinite. They are not growing toward infinity. They already are infinite.

Well that is YOUR mistake.

They are not growing toward infinity.

The sets are growing forever. Infinite growth.

Infinitity is not a number.

Right, it’s a journey. Infinity isn’t something that “just is”. It doesn’t just appear

We are not saying that your two counters will converge on the same number.

I never said nor thought you said that. My argument doesn’t require it.

kairosfocus: Mapou is right to highlight that in our calcs and measurements when we touch down in the real world, we do go to things that are inherently discrete state and non-continuous.

That’s not his claim. Mathematicians and scientists work with the continuum and infinity all the time. Models are never complete or perfectly accurate, but the mathematics of the continuum and infinity have been very powerful tools in mathematics and science.

Pure poppycock. There is no such thing as a mathematics of the continuum and infinity. If it uses numbers, which are discrete by definition, it is discrete, period. Jumping up and down, foaming at the mouth and screaming that your math uses continuity and infinity makes no difference other than you putting your foot in your mouth. And since you insist on referring to yourself as a “we” (are you possessed by demons, goddamnit?), you all have your feet firmly planted in your mouths. LOL.

kairosfocus: For Newton, his original value on earth-moon forces was about 10% as rounded off and in light of available measures.

When you calculate the trajectory of a falling object, per daveS’s comment, we reasonably assume the force acts continuously on the object, and that the object moves continuously through space. We can then solve the differential using mathematics based on the continuum.

You can assume anything you want about bodies moving continuously through space but it’s still complete BS.

To claim that Euclid, Newton, and Einstein were engaging in “crackpottery”, as Mapou does, is not supportable.

They were all crackpots in the things they did not understand. Pathetic crackpots, in fact. I revere only the truth, not some mortal beings who are long dead and buried. If they were so hot and mighty, where are they? Humans die but the truth remains.

A “contrived relationship” is irrelevant for set theory.

Only for this one obviously irrelevant aspect of it. You cannot contrive a relationship to say that one set is a proper subset of another. Do you understand that?

{apples, beans} is not a proper subset of {a, b, c} just cuz I can contrive a relationship using only the first letters of the words in the first set.

So I would say the word is very relevant to set theory. Care to try again?

You claim that philosophy is useless, and yet you have no counter to my argument about how using Cantors method for pairing objects in two infinite sets is completely arbitrary and therefore solves nothing. Once I change the names of different kinds of numbers (rational, real, whole, integers, etc…) to one name, which includes any kind of number, Cantors arguments becomes meaningless. If I no longer have names for all the different kinds of numbers, then one infinite set simply contains varying numbers, and another infinite set also contains varying numbers.

I have no reason to say one set is bigger than another, if I have no separate names for different types of numbers, they are all just numbers. Why would I use his method of bijection, if the names for numbers is just numbers, rather than some man made category for different kinds( There are lots of other ways of categorizing numbers if I insisted on giving them the name one wants, why didn’t he match numbers with zeros to numbers without zeros, its just as arbitrary).

Cantor’s argument thus fails. Philosophy beats math. You, and Zacheriel, and Jerad have no defense for that.

And the journey never ends so there isn’t any “entire journey”.

Correct. Why is it that some people refuse to accept this simple truth?

Claiming that infinity exists is like claiming that there is such a thing as an infinitely smooth circle even though Pi can never be fully computed. Never. The BS we are taught in science is enough to make a grown man cry.

Next, I think we should (–> and yes, there is that pesky OUGHT again; here, duty of care to accuracy, balanced reasonableness and fairness) distinguish the world of abstract concepts and sets, functions and continuous lines or variables, and our finite, discretely observed, measured and computed world.

It is reasonable to apply mathematical models to the world, but we need to be aware of limitations. And indeed the point that we speak of point particles with masses and force fields of inverse square character should serve as a simple first warning on limitations.

(Cf the exchanges here on ways others have pondered this matter, and its enduring subtle significance. Including a pointer to the idea of strings. Mathematical models are not going to be exact physical realities, especially where there is some simplification involved, and/or where elaboration in an educational context will run into deep waters very fast. [This opens up the problem of trusting authorities with simplifications of complex matters, which has become a sore point in a context where crucial areas of science are too often increasingly tainted with ideological agendas. In such cases, appeal to consensus on the one hand may not be good enough, and on the other, we should not get into such a suspicious mindset that we dismiss on a blanket basis without hearing out the deeper explanation or context. In my own experience, I found that telling students that there was a simplification involved [–> E.g. sun and planets atomic model backed up by Rutherford’s experimental findings] and suggesting look-up for those who were going on further, was often enough. Empirically reliable enough models used within zones of tested reliability have their own justification [–> there’s that pesky OUGHT again], but let us not fool ourselves [–> and again] that we have seen all there is to see. Even so “simple” a thing as a mirror implies on laws of reflection a virtual half-cosmos behind it.])

I now suggest that we are finite, fallible, bounded in our rationality, and too often ill-willed.

In that context, we can pause and then recognise that whatever we actually do is bound by our finite space-time existence, and that actual calculation resulting in actual numerical values will indeed always be discrete.

It is algebra and analysis and onwards that allow us to think in terms of a mathematically precise meaning of continuum and the transfinite, as well as the complex domain, vector spaces and more, much more.

And yes, we have now left the concrete, empirical world.

We are in a world of mathematical, abstract conceptual objects that nevertheless have such realities governed by the logic of coherence and systems of articulation that we ever so often find astonishing predictions based on the mathematics.

Let us be grateful for that, and let us ponder that if mental abstractions work so well in the empirical world, maybe they were in fact built in from the outset, i.e. the conceptual-logical world of mathematical forms may well be part of the design architecture of the observed, experienced world.

And of course that means we are back at the observation that the ghost of Plato is laughing, he had a point. Though not necessarily the whole story, as usual.

Where, the continuum does point to the infinitesimal, and so also to the transfinite.

(To my mind, maybe we should go back to thinking in terms of looking at the hyper-reals and their reciprocals, the infinitesimals, as a gateway into Calculus, and onwards wider analysis. I see no reason to ignore a model that has been continually cropping up for over 2,000 years and has been yielding pretty good results. And indeed looking back at old editions of Granville Smith and Longley, I found the use of infinitesimals quite sensible as an introduction back when. The approach of limits, neighbourhoods and the like, is a bit clumsy at that first level, by contrast. And, as basically a physicist, I see no good reason why we should exile time from consideration as we look at space. It is after all x(t), y(t) etc that we really deal with! Trajectory is not suspect, and when we say that the finite experiential process allows us only to point to the infinite that is a way of saying that we have jumped to the abstract world, and why not call them forms and be done?)

Plato’s parable of the cave — 2400 years on, pace dismissiveness to phil — still has bite, even in terms of the difference between experienced worlds of mundane existence, and the abstract space of idealised concepts.

Which can have their own surprising existential powers.

I find it intriguing that one cannot fold a paper clip into a square circle, because the core properties of the two elements stand in mutual contradiction.

That is an astonishing point, if one thinks about it.

A logical constraint can lock out realities that we might otherwise think possible. (And is it just me who notices how many waste paper baskets are done as circles transitioned to squares or vice versa? Or more properly, approximations . . . )

Our world is full of clues that point beyond itself.

An entirely appropriate point on a day when we celebrate one whose birth was heralded by “Angels we have heard on high . . . ”

And, the same to you. De beaches are nice, much nicer dan ice . . .

(But, how often have I parked right next to a beach, only to be heading in the opposite direction to work. Not to wade, swim or fish . . . and there’s a new year resolution for you!)

Calculus as invented by Newton and Liebniz works because of the continuous nature of mathematical functions. For a given modelled real life situation the math of continuity works and gives us answers that can be measured on the ground.

If it was all just hogwash it would have been dropped decades if not centuries ago. It is some of the powerful,useful and dare I say beautiful analytic tools invented by human beings. And it works. It helps you solve real world problems. And it rests on continuity.

Take a class in complex analysis (that’s calculus with imaginary numbers). Talk about falling down the rabbit’s hole! And there are good, solid, real world applications. I couldn’t believe it but it works. And it’s used. By engineers every day.

Fourier analysis is an amazing and powerful technique, again based on continuous functions. I’ve had engineers tell me, whatever else you teach make sure they understand what goes into Fourier analysis. It works.

The mathematicians who developed Analysis (the area of math that includes Calculus) were generally working on physical problems. And they had (and continually have) other mathematicians scrutinise and criticise their work. Even as an undergraduate no holds are barred. You get something wrong, you’re told. Unless you’ve been through it it’s hard to convey the impossibility of getting something accepted that isn’t true or doesn’t work.

Merry Christmas, Mung. I hope Santa brings you less confusion this year.

Merry Christmas keiths.

Why OUGHT Santa bring me anything?

Why OUGHT Santa bring me “less confusion”?

Wny OUGHT anyone take your subjective morality seriously?

Why is it that in your subjective moral announcements you presume that everyone OUGHT TO agree with your subjective moral announcement, as if it were an objective truth?

Calculus as invented by Newton and Liebniz works because of the continuous nature of mathematical functions. For a given modelled real life situation the math of continuity works and gives us answers that can be measured on the ground.

If it was all just hogwash it would have been dropped decades if not centuries ago. It is some of the powerful,useful and dare I say beautiful analytic tools invented by human beings. And it works. It helps you solve real world problems. And it rests on continuity.

Amazing. Nobody here said that calculus was hogwash. You are blatantly lying in order to make a stupid point and you know why, Jerad? It’s because you are a jackass. If calculus used continuity as you claim, why is it that you cannot use it to create a perfect circle, huh? Show us your infinitely smooth circle or shut the hell up.

Show us a reason that we should continue to argue with jackasses.

Fourier analysis is an amazing and powerful technique, again based on continuous functions. I’ve had engineers tell me, whatever else you teach make sure they understand what goes into Fourier analysis. It works.

I use Fast Fourier Transforms (FFT) in my work in speech recognition and I can assure you there is nothing continuous about it. It’s 100% discrete.

Gives you a continuous circle. And it's differentiable.

e^(it) gives you a circle in the complex plane.

y = sqrt(r^2 – x^2). Gives you the top half of a circle radius r.

All of these are continuous functions. You can zoom in as far as you like and the curves stay smooth.

In polar coordinates it's dead easy: r = whatever radius you want. And you can also do calculus with that.

You don't have to argue with me at all. I'm just telling you they way it is. Most of the theorems of calculus depend on continuous functions. If the theorems weren't true then calculus would be on pretty shaky ground. You can link them up if you don't believe me.

it’s Christmas Day, let’s ease back several notches. In addition, I suggest the only place where we have perfect circles is an ideal world of mathematical forms. Once we hit concrete reality, we are dealing with messy approximations and round-offs. And, I am glad to see strong affirmation of inherently mental realities that speak with power into our material world.

Here is Plato’s ghost, laughing as he opened the door for theism by way of perceiving design . . . ironically, an act of mind:, The Laws, Bk X

Ath. . . . when one thing changes another, and that another, of such will there be any primary changing element? How can a thing which is moved by another ever be the beginning of change? Impossible. But when the self-moved changes other, and that again other, and thus thousands upon tens of thousands of bodies are set in motion, must not the beginning of all this motion be the change of the self-moving principle? . . . . self-motion being the origin of all motions, and the first which arises among things at rest as well as among things in motion, is the eldest and mightiest principle of change, and that which is changed by another and yet moves other is second.

[[ . . . .]

Ath. If we were to see this power existing in any earthy, watery, or fiery substance, simple or compound-how should we describe it?

Cle. You mean to ask whether we should call such a self-moving power life?

Ath. I do.

Cle. Certainly we should.

Ath. And when we see soul in anything, must we not do the same-must we not admit that this is life?

[[ . . . . ]

Cle. You mean to say that the essence which is defined as the self-moved is the same with that which has the name soul?

Ath. Yes; and if this is true, do we still maintain that there is anything wanting in the proof that the soul is the first origin and moving power of all that is, or has become, or will be, and their contraries, when she has been clearly shown to be the source of change and motion in all things?

Cle. Certainly not; the soul as being the source of motion, has been most satisfactorily shown to be the oldest of all things.

Ath. And is not that motion which is produced in another, by reason of another, but never has any self-moving power at all, being in truth the change of an inanimate body, to be reckoned second, or by any lower number which you may prefer?

Cle. Exactly.

Ath. Then we are right, and speak the most perfect and absolute truth, when we say that the soul is prior to the body, and that the body is second and comes afterwards, and is born to obey the soul, which is the ruler?

[[ . . . . ]

Ath. If, my friend, we say that the whole path and movement of heaven, and of all that is therein, is by nature akin to the movement and revolution and calculation of mind, and proceeds by kindred laws, then, as is plain, we must say that the best soul takes care of the world and guides it along the good path. [[Plato here explicitly sets up an inference to design (by a good soul) from the intelligible order of the cosmos.]

. . . . self-motion being the origin of all motions, and the first which arises among things at rest as well as among things in motion, is the eldest and mightiest principle of change, and that which is changed by another and yet moves other is second…

Perfect example of shooting the breeze.
Merry Christmas to all.

MT: Oh, the irony of your handle! The self-moved agent is the only entity truly free to think for itself. And, an infinite stepwise successive causal chain is an attempted traversal of a countably transfinite set, which fails. Enjoy the day. KF

A self-mover would have to both undergo motion and initiate it, thus combining actuality and potentiality with respect to the same thing within itself, which is impossible. It would be ‘transported as a whole, and transport itself with the same motion, being one and indivisible in form, and be altered and alter, so that it would teach and learn at the same time, and heal and be healed with the same health’. Aristotle’s alternative is to consider a self-moving thing as a composite of unmoved and moved parts, rather than a whole moving itself as a whole

We must begin our examination with motion, for surely it is not only false that the essence of soul is correctly described by those who say that it is what moves (or is capable of moving) itself, but it is an impossibility that motion should be even an attribute of it. That there is no necessity that what originates motion should itself be moved has been said before. (De an. I.3.405b31-406a6)

the placement of one Penrose tile can affect things thousands of tiles away—local constraints create global constraints. But if a crystal forms atom by atom, there should be no natural law that would allow for the kind of restrictions inherent to Penrose tiles.

It turns out crystals don’t always form atom-by-atom. “In very complex intermetallic compounds, the units are huge. It’s not local,” says Shechtman. When large chunks of crystal form at once, rather than through gradual atom accretion, atoms that are far apart can affect one another’s position, exactly as do Penrose tiles.

end quote:

I wonder if anything else in nature forms all at once rather than through gradual algorithmic steps?

LoL!@ Roy- All of those words have meanings- accepted and standard meanings- in set theory. OTOH contrive and derive remain the same and I have explained it all. Strange that people ignore my explanations and prattle on as if their blatherings mean something.

we went from shooting the breeze about infinite non-repeating tiles to naturally occurring quasicrystals to squishy practical self-assembling organics at the center of life in less than 40 years

I have no idea what you are saying. How does the concept of binary digits help with the problem of a totally subjective method for creating bijections of two sets?

I can two sets and think of thousands of ways to match one from one set up with a counterpart in the other. The problem still doesn’t go away. If the premise at the beginning is that each set contains an unlimited amount of elements in the sets, then neither set can be said to contain more or less than the other, they can’t both contain the same amount of elements and different amount of elements at the same time-math can not rescue the logic out of the problem.

The diagonal line game is a canard, that traps people who refuse to think about it very deeply, but instead prefer to believe whatever they are told.

The abstract mental plane on which the tiles are arranged does not exist in the phyiscal universe it exists outside the cave.

Where is philosophy in tiles arrangement on a plane ? It is geometric arrangement. Plane is a geometric structure. It is not ‘outside’ Plato’s cave or dugout.

I have no idea what you are saying. How does the concept of binary digits help with the problem of a totally subjective method for creating bijections of two sets?

It is what Cantor’s diagonal argument is about (refer Cantor’s 1891 article). It is the diagonal argument using uncountable set.
You can ponder this too : If you stand in between 2 mirrors ( thus creating an infinity mirror), and your friend stands between 2 mirrors of smaller length, are the ‘infinities’ created by your mirror images and your friend’s images same ?

You seem to just being pulling out half concepts, and trying to string together some kind of argument, that I am pretty sure you don’t even know what you mean.

I know what Cantor’s little card game is, but the point is there is no reason to insist on pairing things up the way he does, as if its the only way. Furthermore, its just a silly diagram on a piece of paper, I could place all the numbers with curly shapes on one line, and numbers with straight lines on the other, it is just as meaningful.

And, you do realize that the reflections in mirrors don’t really go one forever, right?

No.You are discussing Cantor’s diagonal argument and I am asking you to refer to his argument which is based on infinite sequence of binary digit set.

but the point is there is no reason to insist on pairing things up the way he does,

Pairing up is a simpler way (than counting elements in set) of seeing if sets are equal.

And, you do realize that the reflections in mirrors don’t really go one forever, right?

You do realize that fading of images is because of not enough lux reaching your eyes? Technically, the mirror is reflecting at geometric progression of 1/n^2, whatever the nth image may be.

MT: Yes, there were debates back then too. Let’s just use a modern set of terms, reflexivity and lag or memory, forming feedback loops. Or, are you unwilling to acknowledge that (a) we influence our own selves across life and also in much shorter frames to milliseconds, and (b) we are responsibly free and self-acting? If the former, that’s a plain breach of common sense to the point that you imply that one cannot acquire a self-formed view, or learn a profession and act in light of the acquired knowledge and skills. If the latter, you undermine learning, knowing, cognition and even the ability to choose to acknowledge force of evidence and reason. KF

fifthmonarchyman: You must find why it went out of date

Never said it went out of date. It just got stale. With regards to infinity, philosophical musings have been supplanted by mathematical proofs.

Mapou: There is no such thing as a mathematics of the continuum and infinity. If it uses numbers, which are discrete by definition, it is discrete, period.

And if it uses continuous functions, then it is continuous by definition.

Zachriel: To claim that Euclid, Newton, and Einstein were engaging in “crackpottery”, as Mapou does, is not supportable.

Mapou: They were all crackpots in the things they did not understand. Pathetic crackpots, in fact.

Eppur si muove.

phoodoo: If I no longer have names for all the different kinds of numbers, then one infinite set simply contains varying numbers, and another infinite set also contains varying numbers.

The naming of sets doesn’t change whether a given set of numbers if countable or not. Take the set {1, 2, 3}, for instance.

phoodoo: I have no reason to say one set is bigger than another, if I have no separate names for different types of numbers, they are all just numbers.

The naming of sets doesn’t change whether a given set of numbers if countable or not. Take the set {cat, trout, dolphin}, for instance.

Mapou: Nobody here said that calculus was hogwash.

“Calculus has historically been called ‘the calculus of infinitesimals’, or ‘infinitesimal calculus’.”

Mapou: If calculus used continuity as you claim, why is it that you cannot use it to create a perfect circle, huh?

it’s Christmas Day, let’s ease back several notches. In addition, I suggest the only place where we have perfect circles is an ideal world of mathematical forms. Once we hit concrete reality, we are dealing with messy approximations and round-offs.

Actually, this ideal world of mathematical forms in which perfect circles exist is an impossible world. Why? It’s because numbers are discrete by definition. So if anything uses numbers (this includes all mathematical functions), it is discrete. So, no matter how continuously smooth you think your circle is, it isn’t. Surprise!

IOW, if Pi cannot be written down, all possible circles are discrete. It’s not rocket science, really.

Mapou: So if anything uses numbers (this includes all mathematical functions), it is discrete.

There are many continuous mathematical functions.

Funny, I have not seen one yet. Calling something continuous does not make it so. I could say that your cranium is infinitesimally small, too, you know. In your case, it might be true. Let’s see. How many demons can fit in an infinitessimally small cranium? LOL.

Mapou: if Pi cannot be written down, all possible circles are discrete.

Pi.

At least one of your demons has a sense of humor. But sooner or later, the day of unspeakable torment will be upon you all.

The functions I listed are all continuous over their domains. And not just because i said so, because they meet the definition. Look it up.

Sin(x), cos(x), 2^x are continuous for all real values of x. Log(x), sqrt(x) are continuous ober their domains. Again, because they meet the criteria. The Wikipedia page is as good a place to start as any but one way to think about is: you can zoom indefinitely far over the domain and the graph will not get jagged.

Fractals are nowhere continuous. I think. I’ll have to look that up later.

As I said, the definitions and examples can be easily found. Any basic calculus book will have a good discussion of the topic and will explain why continuous functions are so important in calculus.

A set of infinite numbers is already uncountable! Since both sets contain un-countabilities, your point has no meaning. So you have just made me realize you don’t understand the problem at all. You are just searching for answers online.

Its kind of disappointing, because I then know that any counter arguments you are proposing are really just loose threads that aren’t based on a real contemplation of the issue. Like when you say there is no one to one correspondence, I could try to draw this point out and ask what you mean by a correspondence, but since you don’t know what it means, I guess we will gain nothing.

A one-to-one correspondence means you can show no element is left unmatched to a member of the other set.

For example: here’s a one-to-one correspondence between the positive evens and the positive odds

1 2
3 4
5 6
Etc

If you specify an element in either set I can tell you what element in the other set it’s marched with. No element is left out it’s a one-to-one correspondence so the sets are the same size.

The point is not that you can’t zoom in till you turn green. The point is that, regardless of how far you zoom in, it is still discrete numbers. It’s always discrete. Deny until you pass out.

You know, Parmenides and his disciple Zeno pointed out the stupidity of infinite series ages ago. You bozos still have not learned the lesson. No wonder physics is filled with braindead monstrosities like time travel, infinite parallel universes and other crap. Intellectual incest is alive and kicking in science.

Wiki: In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set.

You can keep zooming in without limit, there is no smallest interval. And all numbers can be written with infinite decimal expansions. Some are nice enough not to have a pattern and you can still write some of them down.

One solution to x^2 = 2 is the sqrt(2). Exactly.

The ratio of the circumference of a circle to its diameter is pi. Exactly.

There are an infinity of numbers between any two numbers you give me on the number line. That matches the mathematical definition of continuous.

The concept of continuity has proved very powerful in mathematcs and it has several equivalent dedinitions. You may think it’s a lie but without it some of the theorems supporting FFT would not be true. You can still grind out the calculations based on someone else’s algorithm but it doesn’t mean continuity doesn’t underlie the whole endeavour.

Jerad, I am still waiting for you to explain how something that uses numbers (the epitome of discrete entities) can be non-discrete. And I’m still waiting for you to explain how something that can never be fully counted by definition makes any sense in reality. But then again, forget it. I know you and that demon-possessed ZacKy-O meister are weavers of lies and deceptions. You can always come up with another cockamamie non-answer.

Mapou, if you really find much of mathematics equivalent to voodoo then I guess we’ve little to talk about. I’ve been trying, in my imperfect and limited way, to explain some things but your view has little in common with the foundations and theorems of calculus. I don’t mind but you calling it lies and hogwash is a bit silly really. Still, it’s your opinion. But there’s a couple hundred years or more of mathematicians that disagree with you

Mapou, if you really find much of mathematics equivalent to voodoo then I guess we’ve little to talk about.

You are right that you and I have little to talk about since I love math and I use it all the time in my research. You are a complete bore, Jerad. A jackass.

The same contradiction which lies at the heart of Cantors proof, also lies in the contradiction of calling an infinite set countable. They are simply two ways of saying the same thing. Once you have told me all of the elements of your “countable infinite set” I can then give you another one you didn’t count.

So its not math, or wikipedia which helps you out of the problem, its the problem of bad definitions giving bad results. Cantor was playing a game that seems to solve a paradox, but the paradox is already built into the question, so the paradox can’t be removed. We can’t halve infinity, or double infinity, so we don’t have a starting or an end. What’s the first countable in an infinite set? There isn’t one.

Take your early geometry lessons as an example. We start with the assumption that we have a point in space which is infinitely small, but discrete. We then have an imaginary line, with another point on the end which is also infinitely small. We travel half the distance from one point to the other over and over again, yet we can never reach the other point, why? Because we have used bad definitions. If a point is infinitely small how can we have a point along the line which is halfway? You can’t. Because a discrete point can’t be infinitely small-we have created a contradiction before we even begin to solve the answer. So the contradiction can never be removed from the problem. All answers will necessitate a paradox.

Same thing with Cantor, the definition of a countable infinite set is a contradiction, there will always be another you didn’t count, just like in the set of real numbers. Infinitely large is the same as infinitely small, there is no beginning and no end.

Once you have told me all of the elements of your “countable infinite set” I can then give you another one you didn’t count.

Here are all all the natural numbers: N.
What natural number did I not “count” ??

So its not math, or wikipedia which helps you out of the problem, its the problem of bad definitions giving bad results.

the definition of a countable infinite set is a contradiction

No, there’s no contradiction. You’re just mixing up words. If you don’t like the definition, make up your own and call it “phoodoo countable” otherwise people will misunderstand you.

Countably infinite means a set lines up, one for one, with the counting numbers. It doesn’t mean you count them all which you could only do if the set was finite.

Sometimes it’s helpful to take stock of where we are at. There seems to be three positions on infinity here.

1) Infinity does not really exist because it is not part of the phyiscal world and when explored deeply leads to paradoxes and is useless in mathematics.

2) Infinity is useful and necessary in mathematics but it’s a waste of time to think too deeply about it.

3) Infinity is useful and necessary in mathematics despite leading to paradoxes and not existing in the phyiscal world. This fact points to a real and profound existence beyond and above the material universe.

There have been a lot of nuts in mathematics and physics who suffered from some kind of neurological disorder such as bipolar disorder or autism. Cantor, Goedel, Einstein and many others were all nuts. This does not mean that they had nothing interesting or worthwhile to offer. But it is a sure bet that their mental illnesses are reflected in their works.

Only a total nut like Cantor would insist that two sets that can never be fully counted to establish their sizes can be compared to determine which is greater.

Only a total nut like Einstein would insist that a body can move in spacetime when anybody with a modicum of logic can see that it’s crap.

Only nuts insist that there is an infinite number of points on a line.

And I say this even though I, too, am a nut. I say it because it takes one to know one. 😀

PS. Don’t get me started on that hopeless lunatic, Goedel.

Countably infinite means a set lines up, one for one, with the counting numbers. It doesn’t mean you count them all which you could only do if the set was finite.

That is precisely the point. If you can’t count them all, the set does not exist.

LoL!@ Roy- All of those words have meanings- accepted and standard meanings- in set theory.

That’s right, Joe! All those words – tree, morass, cardinal – have accepted meanings in set theory. But their meanings within set theory are completely different to their meanings outside set theory.

Which means that the comment of yours to which I was replying:

I have told you that the meanings of words don’t change just because we are using them to describe what is happening in Set Theory.

One of the nice rewards of rejecting nutty sciences like infinite sets and the like is that, once you do, all the paradoxes disappear in one fell swoop. For example, once you reject the silly notion of continuity, all the paradoxes of Euclidean geometry (e.g., parallel lines that meet) simply disappear. Poof. It’s like having a huge load instantly knocked off your shoulders.

all the paradoxes of Euclidean geometry (e.g., parallel lines that meet) simply disappear.

If you mean parallel lines meet in projective plane, then yes they do, because all three types of conics (formed by the lines) are just a cone’s section in projective plane (For Eg- when you look at a painting).

all the paradoxes of Euclidean geometry (e.g., parallel lines that meet) simply disappear.

If you mean parallel lines meet in projective plane, then yes they do, because all three types of conics (formed by the lines) are just a cone’s section in projective plane (For Eg- when you look at a painting).

Yes we are saying some infinite sets are ‘bigger’, have a larger cardinality than other infinite sets. When the idea was first proposed it was generally looked on with scorn by mathematicians but now it’s accepted and understood and used.

We are NOT saying an infinite number can be counted. Infinity is not a number and besides, if it were, you could always add one and get a larger one. Some infinite sets are said to be countably infinite because they are ‘the same size’ as the counting numbers (also referred to as the natural numbers or the positive integers).

But none of this has anything to do with materialism or evolutionary theory. Why would you lump them together? Because you disagree with them all?

I’ve known some very theological mathematicians who would take great offence at your bias. They might even say that as God is infinite it’s inspiring to study the infinite and to find that there too is great beauty and complexity.

But I guess you disagree. Your call. You’re missing some wondrous things, great beauty and great mystery. Fascinating.

Spare me from your foolishness…. if its infinite it is infinite, there is no bigger or smaller…. OK maybe in your imagination. Adding one to infinite does not make it larger because you still can’t calculate it damn….

Infinite + 1 = ?

Infinite – 1 = ?

INFINITE…….

Please don’t patronize me, God is sitting this one out. Secondly it has everything to do with materialists and Darwinists because you lot act in such a way that does not fit the real universe. Prove me wrong, I dare you!

I am not trying to patronise you. KF knows what I am talking about so you can ask him. I’m just trying to explain.

If the universe was designed then surely when we see sets continuing on to infinity we should start down that path?

Infinity plus or minus one is infinity, surely. 🙂

Really though, there is a very elegant proof that there are more real numbers than rational numbers. Nothing to do with materialism. Not a thing. It’s part of mathematics, look it up.

There is another view, that we must accept that inherently ideational things can hold an abstract reality that constrains what is possible and even actual. For instance, there are no square circles as there is a logical contradiction of core attributes. So neither God nor us can bend a paper clip into a square circle. Such impossible beings cannot exist.

Logic — as abstract a thing as we get — constrains reality, and so holds reality in some form. (Which is a big clue. And no, I am not playing at platonism, I am highlighting that the evidence we have points to the fundamental power of ideas in reality, and as ideas seem inextricable from minds, to mind as a foundational aspect of reality. Mathematics, the Achilles’ heel of scientific-technical materialism.)

But, we can start with things such as how natural numbers are necessary beings, and that in a transfinite succession. More or less following a path trod by von Neumann (almost, as usual):

{} –> 0
{0} –> 1
{0, 1} –> 2
{0, 1, 2} –> 3
. . .

OMEGA . . .

EPSILON-NOUGHT

. . .

Strictly, ordinals so far, to get to cardinals, toss away the successor pattern, and for the nth in the sequence from 0, the cardinality of the number is effectively n – 1. That allows us to identify Aleph-null as holding cardinality of the set of natural numbers.

Note those pesky ellipses, we are pointing to an in-principle, a supertask we cannot actually complete, but can contemplate logically — hint, hint on the contemplative, rationally envisioning mind. And, with a suitable set-builder procedure, we can appreciate that the natural numbers cannot not exist, appearing as a direct consequence of a successor process applied to the empty set and a cardinality assignment operation.

Individual naturals are necessary and the set, which is patently transfinite, is also necessary, it cannot not exist in any possible world. Basic Arithmetic operations follow, per logic and things like 2 + 3 = 5 are necessarily true and in simple cases are self evident even to finite, fallible creatures such as we are.

We can take the number line as a useful construction, and look at the interval, [0, 1). We may define a proper fraction as a ratio of two natural numbers p:q, with q > a, interpolating in the interval. We may then transfer to the place value notation system, and might as well use base 10. Thus we see rationals as WHOLE + FRACTION, (partly) filling the gaps between successive numbers, 0, 1, 2, . . .

If we extend by allowing a series, and the usual notation, whole being W and fraction = B, W + B –> W.B, i.e.:

W + b1/10 + b2/100 + b3/1000 + . . .

With W = . . . w3 X 1,000 + w2 x 100 + w1 x 1, any number on the line can be expressed

. . . w3w2w1.b1b2b3 . . .

where, the ellipses indicate infinite series. Countably infinite, i.e. we apply as many digits as are in a set of cardinality Aleph-null. (In many cases we will have a huge array of leading and trailing zeroes. Which, from grade school on we usually ignore.)

It can be argued that for any given W.B1 and W.B2, we may interpolate another number, for convenience, the average of the two, or some intermediate at any rate. Thus, again in principle, we fill in the spaces, so that gaps between the rational numbers are covered. That’s where irrationals and transcendentals lurk, and once we identify any given one, we can arguably show that each is a root of at least a countably infinite set of others, by applying multiples and fractions etc. pi, 2*pi, 3*pi etc and pi/2, pi/3 etc.

So, we have a case of our finitude and pointing to what lies beyond finitude. The numbers we can actually directly handle and compute out to actual expression form a fine dust on the real number line, but the line is continuous per the concept of interpolation. And, extends without limit.

All of this is abstract, and in fact we have introduced continuous variables, i.e. W.B is a continuous real variable quantity. Toss in additive inverses headed the opposite direction and negatives are there too. Pay off a debt – d by depositing d, and you owe 0. That’s where these first were recognised in our civilisation.

This can be assigned, x if we please. To get y, I suggest the complex number approach where i*i*x = – x and plausibly i*x –> y, an orthogonal axis. This allows a 2-d space, with angles defined algebraically by Z = x1 + i*x2, with r^2 = x^2 + y^2 and tan h = x2/x1, etc. We can extend to the ijk system for 3-d space and by viewing t as time, get to trajectories x(t), y(t) etc.

Physics can be added, and so forth.

But the point is, we have an abstract set of concepts with logical constraints implicit that apply to physical reality. And, this brings in the in-principles of those ellipses. Continuous, abstract variables in a mathematical world that models the physical one we experience, with a gateway from the logic to constrain it. And with the transfinite there, at both ends, the infinitesimal and the ultra-large.

I hardly need to say, such has been highly successful in sci-tech and more mundane pursuits alike.

But at the root of it is a ghost, the suggestion that mind sits at the root of reality, that the ideal forms we have discussed insofar as the finite and fallible can (courtesy ellipses) are inherently mental and eternal. Where, had there ever been utter non-being, that would forever obtain as non-being has no causal power.

Eternal mind, eternally holding and contemplating such abstract realities.

The ghost of Augustine is laughing.

And, while I can appreciate why many are concerned on issues of crack-pottery, last I checked, all of us are cracked pots, and there is One who specialises in that:

2 Cor 4:7 But we have this treasure in jars of clay, to show that the surpassing power belongs to God and not to us. 8 We are afflicted in every way, but not crushed; perplexed, but not driven to despair; 9 persecuted, but not forsaken; struck down, but not destroyed . . .

The glory from beyond shines out through the cracks . . .

MT: If you mean parallel lines meet in projective plane, then yes they do, because all three types of conics (formed by the lines) are just a cone’s section in projective plane (For Eg- when you look at a painting).

Mapou : Only if you assume continuity which is hogwash.

If you don’t assume continuity, then a line is no longer a line -it is a line segment.

Andre, I understand the tendency to suspect authority (especially ideologised authority) in our day. And, we have been burned by the hot stove a few times. But our response should not be as Mark Twain’s cat, which would never sit on a stove again thereafter — hot or cold. There is much to object to, and the issues of the strange difficulties of the transfinite lurk, but that should not lead us to hyper-skeptical dismissiveness. At minimum, let us respect serious, empirically highly reliable work, even if we have reservations on points that look doubtful. Yes, the abstract continuum is riddled with the transfinite, but so are ordinary counting numbers and rationals; things that we can only contemplate with ellipses. Yes, there are horrors there, which led the ancients to erect forbidden zones. But, with reasonable processes — and yes, Cantor paid an awful personal price — we can get a few glimpses that allow us to have a higher confidence in our work than otherwise. BTW, that was also how I came to swallow the issues of quantum and relativity, green eggs and ham stuff that were pretty hard to deal with. KF

MT, any finite line segment is a continuum, with transfinite cardinality. We contemplate here, the shocking issues that lurk in so simple an exercise as addressing a line between points. And, back in the day, the debate on how many angels can dance on the head or point of a pin was about location vs extension, i.e. the infinitesimal contemplated by defining a point as pure location sans extension. The world is stranger than we imagine, perhaps stranger than we CAN imagine, and I don’t remember who I first saw with that one. KF

pardon but kindly look at 351, on the uncounted, uncountable numbers, the irrationals and transcendentals, with pi and e as familiar cases, a lot of logarithms and trig vales being suspected cases, and more. They probably outnumber the rationals and integers.

Some argue as 2^Aleph Null to Aleph null or thereabouts. (There is a debate as to the relationship between c the continuum number and the successive power set scales on aleph null.)

Wiki:

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are pi and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x^2 – 2 = 0 . . . . In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a new method for constructing transcendental numbers.[7] In 1878, Cantor published a construction that proves there are as many transcendental numbers as there are real numbers.[8] Cantor’s work established the ubiquity of transcendental numbers.

Jerad, feel free to cite. And the solution to a freezing New Year’s is as close to hand these days as your friendly travel agent, but of course even with Brent just now at $ 60.24 /bbl, that’s not cheap. Funny how $60/bbl looks cheaper going down than it did going up, but then the inflationary impact of the past several years should not be underestimated. I only hope we have an era of fairly cheap energy ahead, I think the Saudis only have so much influence and face the issue that shale and fracking stare them in the face. Oh for LIFTR-Th and real fusion. KF

Too little money and family responsibilities preclude travelling I’m afraid but I would very much like to visit the Caribbean some day. I’ve seen pictures, clear, warm water to die for. Sigh.

Jerad, I understand. Many’s the day I have had to park next to just such a beach, then glance wistfully at the waves, and turn away to deal with the issues of the day for a client or two. KF

Mapou: Calling something continuous does not make it so.

The continuum is a mathematical model. Between any two distinct points on a line, there is a point in between, and there are no gaps (least upper bound property).

phoodoo: A set of infinite numbers is already uncountable!

In mathematics, countable is defined as having a one-to-one correspondence to the natural numbers or to a subset of the natural numbers.

Mapou: The point is that, regardless of how far you zoom in, it is still discrete numbers.

The Euclidean plane and Newtonian manifold are continuous by mathematical definition.

phoodoo: Once you have told me all of the elements of your “countable infinite set” I can then give you another one you didn’t count.

Consider the set of natural numbers.

fifthmonarchyman: 3) Infinity is useful and necessary in mathematics despite leading to paradoxes and not existing in the phyiscal world. This fact points to a real and profound existence beyond and above the material universe.

What paradoxes?

Mapou: If you can’t count them all, the set does not exist.

In set theory, there is the set of natural numbers.

Mapou: all the paradoxes of Euclidean geometry (e.g., parallel lines that meet) simply disappear

Parallel lines never meet in Euclidean geometry.

phoodoo: So there are an infinite number of infinities in your set N?

That is correct. If you divide N into subsets, at least one of them must be countably infinite. Indeed, there are uncountably many countable subsets of N.

Mapou: Only if you assume continuity

Assume there is an additive identity.

Andre: if its infinite it is infinite, there is no bigger or smaller

Cantor proved that the real numbers are a larger cardinality than the counting numbers.

Me_Think: If you don’t assume continuity, then a line is no longer a line -it is a line segment.

Actually, there is a one-to-one mapping between the line and the line segment, or the line and Euclidean space, or if you prefer, the lotus blossom and the cosmos.

phoodoo: Are there or are there not an infinite number of infinities in the set of all natural numbers?

If you divide N into subsets, at least one of them must be countably infinite. Indeed, there are uncountably many subsets of N.

Two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set.

For example let A = {1,2,3,4,…}
Let B= {2,4,6,8,…}

We can remove every element of set B from set A and still have an infinite set left. So how can the two sets have the same number of elements?

phoodoo: And how large is the subset that contains 1/4 of all natural numbers? Infinite

Countably infinite.

phoodoo: And how large is the subset of the subset which contains half of all natural numbers? Infinite

Countably infinite.

phoodoo: And how large is the subset, 1/1000 of the subset that contains half of all the natural numbers? Infinite.

Countably infinite.

phoodoo: So we have just confirmed that the set of natural numbers contains an infinite number of infinities.

Not just infinite, but uncountably infinite. Per Cantor’s theorem, the cardinality of the power set of A (the set of all subsets of A) is greater than the cardinality of A.

phoodoo: you believe anything someone with a title tells you.

We can remove every element of set B from set A and still have an infinite set left. So how can the two sets have the same number of elements?

That’s the way countably infinite sets work. The positive integers (countably infinite) take away the evens (also countably infinite) leaves you the odds (also countably infinite).

It takes a while to get used to it. But, like relativity and quantum mechanics, it starts to make sense eventually.

That’s the way countably infinite sets work. The positive integers (countably infinite) take away the evens (also countably infinite) leaves you the odds (also countably infinite).

Then the two sets do NOT have the same number of elements. Thank you.

So we have just confirmed that the set of natural numbers contains an infinite number of infinities.

I thought you meant that in a different way. And many of ‘infinities’ you are thinking of overlap. I suppose what you say is right although i wouldn’t say it the same way.

But yes there are a lot of countably infinite subsets of countably infinite sets. Again, the criteria for countably infinite is: can it be matched 1-to-1 with the natural numbers.

can it be matched 1-to-1 with the natural numbers.

That’s the trick, isn’t it?

Two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set. And that also means there isn’t any real 1-to-1 correspondence.

Indeed, you are not incorrect, but on a serious note, and I stand by this, materialists, do not act in a way that fits the real universe.

Darwinism, multiverse, and this pile of junk we’ve been dealing with in this thread. Sure the mind can imagine all kinds of things, but that does not make it so.

You can put the positive integers and the positive even inters into a 1-to-1 correspondence so, even though one set contains the other and has more elements they both have the same cardinality. All of the following are countably infinite:

The integers, the positive integers, the multiples of 2, the multiples of 3, the multiples of any counting number, the perfect squares, the perfect cubes, the primes (not sure about the prime pairs yet), the rational numbers (all the possible ratios of integers). All if these sets are the same ‘size’, the same cardinality. It is weird to think about at first. But all of the above sets can be put into a numbered list which will leave none of the set out. And the numbering puts the set into a 1-to-1 correspondence with the counting numbers.

I’m sorry if you don’t like it or think it’s rubbish. But it’s established, accepted and it works.

You can put the positive integers and the positive even inters into a 1-to-1 correspondence so

Not really and I have explained why. That you have to ignore that explanation tells us that you cannot deal with it.

even though one set contains the other and has more elements they both have the same cardinality.

So cardinality doesn’t refer to the number of elements in a set? I was always taught that the cardinality of a set is the measure of how many elements it contains.

But it’s established, accepted and it works.

It isn’t used for anything, Jerad. So how can you say that it works? Perhaps you mean it works for lazy people who don’t want to think about it and want and easy way out of the obvious contradictions.

And by “established” Jerad means “accepted by those who are too afraid to stand up and be counted”

Can you write down a pattern for your infinite set so you’re sure that following the patter nothing will be left out?

Here’s a sequence: 2, 5, 8, 11, 14 . . .

The pattern is: add three to get the next term.

Here’s my 1-to-1 correspondence with the counting numbers . . .

The counting number n is matched with 3n – 1.

n = 1 is matched with 2
n = 2 is matched with 5
And so on.

You can give me a counting number n and I can tell you what term of the sequence it’s matched with (n = 23, for example is matched with 68). Or you could give me a term in my sequence and I can tell you what counting number gets matched with it (110 is in my sequence and it’s matched with the counting number 37).

So now essentially I’ve got the elements of the two sets paired and no element of either set is excluded. The sets must be the same size or some element of one set would be without a partner.

No matter how many aspersions you choose to cast you will find what I’m telling you in many textbooks and web pages. Please go have a look. It is not a lazy way out and there are millions of people who are standing up and being counted as supporters, KF among them.

And thank you for continuing to avoid my arguments. That alone is very telling.

I am trying hard to answer your questions even though they are stated in non-standard forms.

And that’ve other set tells me the difference in size between the two sets. Not my fault that you are stuck in a box.

Except it’s just formula relating one element to another. What does that tell you about their relative sizes? Assuming you’ll say one set is three times the size of another how can you prove that? My formula shows that every element of each set is married to an element of the other set. No element is unmarried. How could that be if one set was larger than the other?

Find me an element of one set that is not married to an element of the other set. That has to be true if one set is bigger than the other set

This is where your approach fails. You cannot pick an element in one of the sets that is not married to an element in the other set. There are no bachelors and no polygamous elements eiher. The sets are matched up, one for one. How can that happen if they aren’t the same size??

Just find me an element of one set that isn’t married, that’s how you falsify my correspondence. That’s all you have to do.

Except it’s just formula relating one element to another.

Heh.

My formula shows that every element of each set is married to an element of the other set.

Heh. I say your formula shows the difference in sizes between the two sets

Assuming you’ll say one set is three times the size of another how can you prove that?

The formula. And it’s 3 times minus 1 in the scenario you are referring to.

You cannot pick an element in one of the sets that is not married to an element in the other set.

And yet I have. Here it is, AGAIN:

Two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set. And that also means there isn’t any real 1-to-1 correspondence.

Set 1= {1, 2, 3, 4 . . .} Set 2 = {2, 5, 8, 11 . . . } I marry all the elements of the first set with elements of the second using the formula

n from the first set marries 3n – 1 from the second set

If one set is bigger than the other then you should be able to find an element of the bigger set that is not married to an element from the smaller set.

Can you do that? If you can’t then how can the sets be different sizes?

Your assertion is meaningless if you can’t find an unmatched element. You say there must be one at least. Name one.

I think I’ve found a 1-to-1 matching that leaves no element unmatched. Prove me wrong.. Find an unmatched element and prove one set is bigger than the other.. Just one element will do.

Prove me wrong Joe. Find an unmatched element from one of the sets using my matching rule. If you can then my 1-to1 correspondence is wrong and one set must be bigger than the other.

We are arguing with demon possessed lunatics on the one hand and dishonest jackasses on the other. I hate repeating myself but here goes.

There have been a lot of nuts in mathematics and physics who suffered from some kind of neurological disorder such as bipolar disorder or autism. Cantor, Goedel, Einstein and many others were all nuts. This does not mean that they had nothing interesting or worthwhile to offer. But it is a sure bet that their mental illnesses are reflected in their works.

Only a total nut like Cantor would insist that two sets that can never be fully counted to establish their sizes can be compared to determine which is greater.

Only a total nut like Einstein would insist that a body can move in spacetime when anybody with a modicum of logic can see that it’s crap.

Only nuts insist that there is an infinite number of points on a line.

And I say this even though I, too, am a nut. I say it because it takes one to know one. 😀

PS. And don’t get me started on that hopeless lunatic, Goedel.

Once again, one cannot compare two sets to determine which one is bigger unless one determine their sizes. Since nobody can determine the size of an infinite set, saying that infinite set A is bigger or smaller than infinite set B is pure hogwash from a deranged mind. Somebody should go over to Cantor’s grave and defecate on it. Then his followers should be tarred and feathered and paraded in the streets. 😀

Read my post 386 and see if you can find an element of one set that is not married to an element of the other set by my rule.

This is the kind of thing Cantor worked on. He set up similar 1-to-1 correspondences and he couldn’t find an unmatched element. So he figured the sets must be the same size. Infinite but the same size.

Anybody who claims that an infinite set has a size is suffering from some deep mental disability. And we all know Cantor was a fruitcake and so are his moronic followers. 😀

Speaking abstractly, how wide is a point, according to sane people? How many points are there on a line segment that is exactly one inch long?

Why speak abstractly and why ask me? I am not one of the idiots who claim that there is an infinite number of points on a line. I don’t believe points exist. I believe there are only particles, their properties and their interactions. Everything else is either abstract or voodoo nonsense.

Now, if you want to know how many particles can fit on a 1 inch segment (the abstract distance between two particles), one can easily calculate it by dividing 1 inch by the Planck length. It’s a very huge number. Here I’m assuming that the Planck length is the fundamental discrete unit of distance. There is a good chance that it is not because it was calculated through dimensional analysis and some people are not comfortable with that.

Mapou (sorry for the predictive text typo above) #393

Cantor looked at set matches like what I’ve listed in #386. He realised he couldn’t find mismatched elements in some pairings. He concluded that there were a whole class of sets that were infinitely large but the same size.

And then he found one where he could get a mismatch. Where the pairing could not be made to work. What could he do? He’d found an infinite set that WAS bigger than other infinite sets. There were different infinities.

Why speak abstractly and why ask me? I am not one of the idiots who claim that there is an infinite number of points on a line. I don’t believe points exist. I believe there are only particles, their properties and their interactions. Everything else is either abstract or voodoo nonsense.

And lots heat and very little light has been created here over the meaning of the size of set. Folks that want to
get to grips with the math might want to think about what size actually means for a set. There are (at least) two different concepts that describe a set’s size.

The cardinality of a set is the number of elements, and as others have pointed, it’s possible to show that the set of all integers has the same cardinality as the set of all odd integers.

There is also the density of set, when it considered a subset of some other set. So, when considered as a subset of the set of all integers the set off all odd integers has density of one half. So two sets can have the same cardinality when they considrered as sets in their own right, while having a different density when they are considered as subsets of another set.

These ideas might not be very intuitive, but they are important and powerful ideas in math, and it’s worth spending some time trying to understand them.

Set 1= {1, 2, 3, 4 . . .} Set 2 = {2, 5, 8, 11 . . . } I marry all the elements of the first set with elements of the second using the formula

n from the first set marries 3n – 1 from the second set

If one set is bigger than the other then you should be able to find an element of the bigger set that is not married to an element from the smaller set.

Jerad, if I am questioning the claim that you are marrying all of the elements using that formula then you cannot use that formula as an argument that you are marrying them.

You have no idea how to engage in a debate that challenges the orthodoxy. If the orthodoxy is being challenged then it cannot be used as evidence in support of itself.

Also this: Two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set. And that also means there isn’t any real 1-to-1 correspondence, is a fact and not an assertion.

Mapou (sorry for the predictive text typo above) #393

Cantor looked at set matches like what I’ve listed in #386. He realised he couldn’t find mismatched elements in some pairings. He concluded that there were a whole class of sets that were infinitely large but the same size.

And then he found one where he could get a mismatch. Where the pairing could not be made to work. What could he do? He’d found an infinite set that WAS bigger than other infinite sets. There were different infinities.

I don’t care how the fruitcake redefined terms such as size, bigger and smaller. I don’t give a rat’s asteroid. All I am interested in is the demise of the braindead idea that nature is continuous.

The cardinality of a set is the number of elements, and as others have pointed, it’s possible to show that the set of all integers has the same cardinality as the set of all odd integers.

And it’s possible to show that the set of all integers has more elements than the set of all odd integers. Oops.

There is also the density of set, when it considered a subset of some other set.

What, with a subset the elements have more space between the commas? 🙂

These ideas might not be very intuitive, but they are important and powerful ideas in math, and it’s worth spending some time trying to understand them.

What import and power would that be, exactly? I have been asking for a long time and no one ever answers except to say it is important and I am ignorant. Not very convincing especially given the fact that I have refuted their claims and all they can do is repeat them without addressing the refutations.

My ‘rule’ is my claim. So, yes, I can use it.. Prove my rule wrong!!!

I say: take an element n from {1, 2, 3, 4 . . . }. Marry/match n to 3n – 1 in {2, 5, 8, 11 . . . }

Can you find an element in set 1 that is not matched to an element in set 2 by my rule? If you can then I will agree that set 1 is bigger. That’s it.

Set 1 has all the elements of set 2 and more. So set 1 should be bigger than set 2. So my rule matching set 1 to set 2 shouldn’t work. So you should be able to find an element in set 1 that does NOT get matched with an element in set 2..

All Cantor the fruitcake showed is that if one counts or expands abstract series, some series will grow faster than others. Big effing deal. Who cares? He had nothing new to say about infinity that we did not know.

My ‘rule’ is my claim. So, yes, I can use it.. Prove my rule wrong!!!

I have, using the mathematics of sets. That is by subtracting all the elements of one set from the other we, together, have demonstrated there are elements left in one set but not the other. That means there are elements left unmarried.

And using the same mathematics of sets I have shown that your formula actually shows the difference in size between two sets with infinite elements. I know that upsets you but too bad.

And it’s possible to show that the set of all integers has more elements than the set of all odd integers. Oops.

Let’s just work with the positive integers.

I = {1, 2, 3, 4. . . }

O = {1, 3, 5, 7 . . . }

n is an element in I. n gets matched with 2n – 1 in O.

I claim that this matching shows that I and O are the same size because no element of I or O are unmatched. Give me an element in I or O and I can tell you the unique element of the other set it is matched to. It’s a 1-to-1 correspondence.

If you’re right there should be at least one unmatched element. Find one and I’ll concede.

Jerad, Obviously you have issues that severely limit you, intellectually.

Let’s just work with the positive integers.

I = {1, 2, 3, 4. . . }

O = {1, 3, 5, 7 . . . }

n is an element in I. n gets matched with 2n – 1 in O.

I claim that this matching shows that I and O are the same size because no element of I or O are unmatched. Give me an element in I or O and I can tell you the unique element of the other set it is matched to. It’s a 1-to-1 correspondence.

If you’re right there should be at least one unmatched element. Find one and I’ll concede.

There are an infinite number and using the mathematics of sets we have found them. That is by subtracting all the elements of one set from the other we, together, have demonstrated there are infinite elements left in one set but none the other. That means there are infinite elements left unmarried.

And using the same mathematics of sets I have shown that your formula actually shows the difference in size between two sets with infinite elements. I know that upsets you but too bad.

I have, using the mathematics of sets. That is by subtracting all the elements of one set from the other we, together, have demonstrated there are elements left in one set but not the other. That means there are elements left unmarried.

I’m using a different matching than you. But, if you’re right you should be able to find an element that isn’t matched. If you subtract elements from the second set by my scheme, by taking them out as pairs you don’t get anything left over.

Explain how that happens without claiming that one scheme is ‘better’ .

Show where my scheme fails.

I take n out of the first set and 3n – 1 out of the second set (if we’re
Talking my first example) as a pair. What gets left behind? Find an element that doesn’t get taken out.

But, if you’re right you should be able to find an element that isn’t matched.

There are an infinite number and using the mathematics of sets we have found them. That is by subtracting all the elements of one set from the other we, together, have demonstrated there are infinite elements left in one set but none the other. That means there are infinite elements left unmarried.

And using the same mathematics of sets I have shown that your formula actually shows the difference in size between two sets with infinite elements. I know that upsets you but too bad.

It is possible to show, and has been shown, that the set of positive integers has more elements that the set of positive even integers. If Jerad et al., were correct then that should not be possible.

Once again, one cannot compare two sets to determine which one is bigger unless one determine their sizes.

Set 1: The grains of sand in the Sahara desert
Set 2: The M&Ms in an unopened packet.

Most people would have no difficulty whatsoever deciding which set is bigger without determining their sizes – but mapou thinks it’s not possible.

For a less extreme example, take a box of bolts and a box of washers. You can easily find out which set is larger by threading washers onto bolts until one of the boxes is empty. Mapou thinks this is impossible too.

Once again, one cannot compare two sets to determine which one is bigger unless one determine their sizes.

Set 1: The grains of sand in the Sahara desert
Set 2: The M&Ms in an unopened packet.

Most people would have no difficulty whatsoever deciding which set is bigger without determining their sizes – but mapou thinks it’s not possible.

For a less extreme example, take a box of bolts and a box of washers. You can easily find out which set is larger by threading washers onto bolts until one of the boxes is empty. Mapou thinks this is impossible too.

Not that I expect mapou to admit he is wrong.

Roy

Haysoos Martinez! What is wrong with the mental midgets of the materialist/Darwinist camp? I’m truly running out patience with that clueless bunch. This is like saying that just because I don’t know the exact number of stars in the universe, I cannot conclude that there are more stars than people. There is such a thing as a size estimate, you know. There are logical, well known ways to estimate the size of various finite sets. The problem is that there is no logical ways to estimate the size of infinity. It is ludicrous on the face of it.

I am fast coming to the conclusion that infinity mongers, Darwinists and materialists are all mentally ill. Hell, we all know that Darwin was a fruitcake, just like Cantor, Godel and Einstein. I am arguing with an insane asylum. Lord have mercy!

Can someone explain to me how to determine the size of a set?

Textbooks say by counting the number of members the set has. However, some people measure the distance between the {} and others just make bald declarations.

I am a card-carrying Yin-Yang dualist. I believe that opposites are ONE. There can be no such thing as left without right, up without down, open without closed, no without yes, first without last or beginning without end. If one exists, its opposite also exists. To believe in infinity is to deny the logic of complementarity. The infinite has a beginning but no end. Infinity mongers are a scourge on civilization. They retard the progress of science by centuries if not millenia. They should all be placed in a mental institution for their own protection and the protection of society, and away from kids and animals. 🙂

Haysoos Martinez! What is wrong with the mental midgets of the materialist/Darwinist camp? I’m truly running out patience with that clueless bunch. This is like saying that just because I don’t know the exact number of stars in the universe, I cannot conclude that there are more stars than people.

Yes. It is. But that is a consequence of what you said. If it is ridiculous, you only have yourself to blame.

There is such a thing as a size estimate, you know. There are logical, well known ways to estimate the size of various finite sets.

Yes, I know that. It makes no difference to what you said.

The problem is that there is no logical ways to estimate the size of infinity. It is ludicrous on the face of it.

Yes, estimating the size of infinity is ludicrous. So ludicrous that either every mathematician is a blithering idiot, or you are completely missing the point.

Based on your dismissal of the entirety of infinite set theory as hogwash, and you description of Cantor as a fruitcake, I expect you think every mathematician is a blithering idiot.

But in fact everyone else is laughing at you and Joe.

Can someone explain to me how to determine the size of a set?

Count the number of elements. If you can’t do that, find a rule that allows a one to one mapping to a set of known size. (If you knew how many people where in a room you could establish if the number of chairs was equal to the number of people just by asking them to sit down).

One the examples above is the sets {0,1,2,3,4 …} and {0, 1, 4, 9, 16 …}. It shouldn’t take too long to see there is a function that maps an element from the first set to one and only element in the second set (the square of every element in the first set appears once in the second), if we can map elements between these sets in this way we know they are of the same size (just like the one-to-one realtionshp between bums and seats let us establish to number of seats in the example above).

In fact, the first set is a special case, the set of all natural numbers (+ve integers), and the cardinality of that set is called aleph_null (indeed the square numbers are the first example of a set with this cardinality on the wiki page for this concept).

I don’t care how the fruitcake redefined terms such as size, bigger and smaller. I don’t give a rat’s asteroid. All I am interested in is the demise of the braindead idea that nature is continuous.

I don’t know much about physics, but I do believe that nature is most likely not continuous, FWIW.

But remember that not all mathematics exists to serve physics or the sciences. What is wrong with working with mathematical concepts that don’t correspond to physical reality if they are interesting and/or useful?

For example, what if someone asked you to calculate the integral of sqrt(1 – x^2) between x = 0 and x = 1.

Is it ok to give pi/4 as the answer, or would you insist that only a rational approximation is acceptable?

But, if you’re right you should be able to find an element that isn’t matched.

There are an infinite number and using the mathematics of sets we have found them. That is by subtracting all the elements of one set from the other we, together, have demonstrated there are infinite elements left in one set but none the other. That means there are infinite elements left unmarried.

And using the same mathematics of sets I have shown that your formula actually shows the difference in size between two sets with infinite elements. I know that upsets you but too bad.

Hey, Joe!

Are these two sets the same size?

Set 1: {1,2,3,4,5,…}
Set 2: {-2,-4,-6,-8,-10,…}

How about these?

Set 3: {1,2,3,4,5,…}
Set 4: {1,(1),((1)),(((1))),((((1)))),…}

Or these?

Set 5: {1,2,3,4,5,6,7,…}
Set 6: {1+2,2+2,3+2,4+2,5+2,…}
Set 7: {3,4,5,6,7,…}

And finally:
Set 8: {1,2,3,4,5,…}
Set 9: {(1,1), (2,2), (3,3), (4,4), (5,5),…}
Set 10: {(1+1), (2+2), (3+3), (4+4), (5+5),…}

I really have to agree with Mapou, talk of preposterous infinity conclusions shows the extent to which many materialists are willing to forgo any sense of thought, in the name of believing in a scientific paradigm.

Indeed Cantor was nuts. Perhaps it is one reason why he was unable to see that numbers (which ever kind you want to call them) are creations in man’s mind. As such they don’t have size. They don’t have cardinality. They aren’t limited or unlimited, except in the sense of what any one person can imagine.

Saying that I can put all of my imaginations about one idea in one set, and that imagination is bigger than another imagination that I decide to call something else in another set is smaller, but unlimited, is surely crazy talk.

Why is the set of all real numbers bigger than the set of all M&M’s I can imagine? Its not. Because since the set doesn’t actually exist, except in my mind, I can make the set any size I want. So my M&M set is bigger. And I can even do a mathematical proof just as accurate as Cantors. Just put all of you real numbers into a graph anyway you choose. Now, I reserve the right to create in my mind one kind of M&M that I will call uncountable, and I draw it on the page, with a symbol, outside your graph. Since you have no one to one correspondence to my M&M, my set is bigger.

That is all Cantor has done. And crazy people (mostly materialists it must be said) take this as fact, and are not willing to use their minds for even just a few seconds to see how dumb it is. Its a collective willingness to be told what to think.

One imaginary set is not bigger than another imaginary set, any more than the set of “funny” is bigger than the set of “enlightened”.

I don’t care how the fruitcake redefined terms such as size, bigger and smaller. I don’t give a rat’s asteroid. All I am interested in is the demise of the braindead idea that nature is continuous.

I don’t know much about physics, but I do believe that nature is most likely not continuous, FWIW.

But remember that not all mathematics exists to serve physics or the sciences. What is wrong with working with mathematical concepts that don’t correspond to physical reality if they are interesting and/or useful?

For example, what if someone asked you to calculate the integral of sqrt(1 – x^2) between x = 0 and x = 1.

Is it ok to give pi/4 as the answer, or would you insist that only a rational approximation is acceptable?

I have no problem when fruitcake mathematicians play in their sandboxes. I do have a problem when they want to impose their sandbox philosophy on science. When that happens, we get sandbox monstrosities such as time travel, infinite parallel universes, black holes, continuum physics, the relativity of motion and position and other similar hogwash. I have no problem with using Pi/4 but I do have a problem with BS like division by 0 and sqrt(-1). I do have a problem when some lunatic insists, in or out of the sandbox, that there is such a thing as an infinitely smooth circle or an infinite set. Infinite anything is crap because it violates the principle of complementarity in that it posits the existence of a beginning without an end. This is absurd. Now, if the fruitcakes had properly labeled their sets with names like ‘progressive sequences’ or ‘expanding series’, I probably would have no objection.

Take the probabilistic nature of particle decay for example. Why is it that particles do not have fixed decay intervals? Physicists have no clue as to why that is. And you know why? It’s because they’ve allowed a bunch of fruitcake mathematicians to step out of their sandboxes and play with grownups. That’s why.

I really have to agree with Mapou, talk of preposterous infinity conclusions shows the extent to which many materialists are willing to forgo any sense of thought, in the name of believing in a scientific paradigm.

They are an elitist group of jackasses and fruitcakes who have managed to convince themselves that science belongs to them and that nobody else but them have the right to conduct science.

I do have a problem with BS like division by 0 and sqrt(-1).

Do you realize that Imaginary number [Sqrt(-1)] is fundamental in calculating the wave functions of Quantum particles? Without imaginary number, there is no QM. Division by 0 is used in many complex analysis (‘Complex analysis’ is a branch of mathematics)

I have no problem when fruitcake mathematicians play in their sandboxes. I do have a problem when they want to impose their sandbox philosophy on science. When that happens, we get sandbox monstrosities such as time travel, infinite parallel universes, black holes, continuum physics, the relativity of motion and position and other similar hogwash. I have no problem with using Pi/4 but I do have a problem with BS like division by 0 and sqrt(-1).

Thanks for the reply. For the record, I also have a problem with division by 0!

I’m not sure I understand how you arrived at your position on pi/4, however. How do you even define pi in your system?

It seems that Keiths has balked (chickened) out of answering if he believes in an infinitely small point. Do any other materialists wish to take a stab at this? Does such a thing exist?

Mung, there are indeed good reasons to talk about the size of an infinite set. In particular because some infinite sets are larger than others. There are more reals than natural numbers, for instance.

I am equally offended by how a small group of not very clever thinkers has managed to pull the wool over the eyes of so many others in the general public, to the point that virtually all of the knowledge you can receive online or in print, is really just these weird scientific skeptic agenda, that is masqueraded as some kind of truth.

All these things on youtube, and on so called science shows, or podcasts, or science forums, to the level of National Geographic, or the BBC or major newspapers, they talk about things like Einstein, or evolution, in a way that suggests that they have a clue about the reality of this information, because it has become part of the cultural fabric of society. When Penn Jillete goes on a rant about evolution, invoking the “great minds” of people like Richard Dawkins, he speaks as if he knows he is right about the subject, because all he is ever exposed to is sources he trusts, which tell him its true. If people like Bill Nye, and the BBC or the New York times write articles about how obvious evolution is, well why wouldn’t he just go along and believe this. So then you have people on the Tonight show, or Saturday Night live, or Huffpost talking about all the dumb people who don’t believe evolution, because well, the celebrities even know its true, they read it!

And where did the writers for the BBC and the New York times get their information? Well, its just out there, everywhere. Or from Jerry Coyne, or Neil Shubin, or Danniel Dennet. Did they understand the voracity of the claims, of course not, but so what. Its everywhere in the public, so go ahead, it must be true-everyone knows it.

Are Cantors sets real, of course, its taught in universities, it has to be true. Don’t you know the only people who doubt it are cranks? Just look it up on the internet, there are all kinds of people online telling you they are cranks, why think?

We are a nation of mass media fools, skeptics who absolutely refuse to be skeptical of anything they are told.

If there were more real numbers than numbers you can count with, you could never know it anyway, because you wouldn’t have the capacity (the counting numbers!) to be able to count to check.

“Oh no, we have run out of the ability to say, oh, there is another one, and another one…”

Earth to keith s- anyone who claims that unguided evolution predicts an objective nested hierarchy is mathematically deficient and has no business trying to correct me in anything math related. And anyone who bitches at me about finite vs infinite and then uses finite sets to make a case for the infinite, is a total whack-o.

I made my case, keith s. You can either deal with it or continue to ignore it. Onlookers aren’t the fools that you think or hope they are.

And keith s- please do TRY to keep up. Jerad introduced the “marrying formula” and my claim is that formula provides the relative cardinality difference between an infinite set and all of its proper subsets.

If you want to know the difference in size between your two sets then figure out the “marrying formula”. I have tried to help you with nested hierarchies, which are related to set theory, and you refused my help and continue to bastardize the concept. There is no way I will listen to you wrt set theory. And that goes especially when you blatantly ignore what I say in defense of my claims.

Hi Roy- Guess what? You can use my methodology that you quoted and figure it out for yourself!!!11!11!!!1!!! When comparing negatives to positives just use absolutes to make it easier on yourself.

And keith s- please do TRY to keep up. Jerad introduced the “marrying formula” and my claim is that formula provides the relative cardinality difference between an infinite set and all of its proper subsets.

Joe, you are at sea on this I’m afraid. The cardinality of any infinite subset of the natural numbers is equal to the cardinality of the natural numbers (aleph_null).

I am equally offended by how a small group of not very clever thinkers has managed to pull the wool over the eyes of so many others in the general public, to the point that virtually all of the knowledge you can receive online or in print, is really just these weird scientific skeptic agenda, that is masqueraded as some kind of truth.

You can write a paper and publish to establish your self as the most learned guy. No one is stopping you. Do you have a counter proof for any concepts you want to negate? Publish it. Afraid of peer review? Self Publish. Let the learned majority be the judge.

I do have a problem with BS like division by 0 and sqrt(-1).

Do you realize that Imaginary number [Sqrt(-1)] is fundamental in calculating the wave functions of Quantum particles? Without imaginary number, there is no QM.

I don’t care. It’s obviously a chicken shit kludge since Sqrt(-1) is BS on the face of it. What QM physicists should explain is what causes this so called wave function. They have no clue.

Division by 0 is used in many complex analysis (‘Complex analysis’ is a branch of mathematics)

I’m equally unimpressed. I have always been of the opinion that any phenomenon that requires a lot of weird math to solve is one that physicists are completely clueless about. I would say that 90% of physicists have no idea that nothing can move in spacetime. Hell, when told about it, they immediately deny that it’s true.

The ignorance of physicists is deep. I remember the faster than light neutrino fiasco in 2011 involving a whole slew of highly paid physicists at CERN and elsewhere. It was embarrassing to say the least.

I have no problem when fruitcake mathematicians play in their sandboxes. I do have a problem when they want to impose their sandbox philosophy on science. When that happens, we get sandbox monstrosities such as time travel, infinite parallel universes, black holes, continuum physics, the relativity of motion and position and other similar hogwash. I have no problem with using Pi/4 but I do have a problem with BS like division by 0 and sqrt(-1).

Thanks for the reply. For the record, I also have a problem with division by 0!

I’m not sure I understand how you arrived at your position on pi/4, however. How do you even define pi in your system?

There is nothing wrong with Pi. One should calculate Pi to whatever precision is required by the problem one wants to solve. Pi is the perfect example of the stupidity of continuous structures. Nobody can fully calculate Pi simply because continuity is nonsense. In nature, there is a limit to how smooth a circle (or any curvature) can be and this limit is enforced by the Planck length, i.e., the smallest fundamental distance between two particles.

n in set 1 gets matched with 3n – 1 in set 2. It’s just matching them up in order of appearance.

My matching links 1 in set 1 with 2 in set 2. Take those two out.

Then 2 in set 1 gets matched with 5 in set 2. Takes those two out.

Continue with 3 and 8, then 4 and 11. So far no element has been unmatched.

Continue on . . . I have shown a way to match each element of set 1 with a unique element of set 2. No element is unmatched. Give me an element in either set and I can tell what element in the other set it is matched with. You can’t just say: you can’t use that without quoting a set theory rule which says it’s not allowed.

If the sets are different sizes then there should be at least one unmatched element somewhere. If you want to disprove my scheme all you have to do is find an unmatched element.

If you can’t find an unmatched element then the sets must be the same size.

If you think there is any mathematical reason my scheme is wrong then please give me a mathematical reference clearly stating so.

And keith s- please do TRY to keep up. Jerad introduced the “marrying formula” and my claim is that formula provides the relative cardinality difference between an infinite set and all of its proper subsets.

All my scheme does is match elements of one set with the elements of another set.

You look at the 3n part and think that says some thing about the cardinality. But you’ve got no mathematical references to back up your claims or your objections. What does relative cardinality difference mean anyway? Give a mathematical definition of the term please. And don’t just say the words mean what they always mean, you know that many disciplines use terms differently and in a specific way.

With my scheme no element of either set goes unmatched with an element of the other set. If there were an unmatched element then then sets would be different sizes. Find an unmatched element and you win.

I’m assuming you’re having a hard time finding an unmatched element or you would have presented it by now.

Show me where my scheme fails.. Don’t just say it does this or that, don’t just say it’s contrived without backing up your use of that term with references. You always ask me for references but you’re very reluctant to provide you own in this case.

Again, find an unmatched element, that’s all you have to do..

The exact same logic can be used to compare the set of natural numbers with the set of real numbers, how can you not see that?

What is the first number in the set of real numbers? I will choose to match the number 1 with that. What is your second number? Since you can’t even say what the second number is, the problem isn’t in the number of elements in any one set, the problem is in you defining what elements are in your set, and what elements aren’t. As soon as you can tell me two of the elements, or three of the elements, then I can tell you what the matching number is from my set of natural elements-its as simple as that.

There is nothing that says I have to match up a one in my set with a one in your set. I can just as well match the number 1 in my set, with the number for Pi in your set. Ok, next….

You would have to be a crazy person (Cantor) to not see how easy this idea is.

Darwinism, multiverse, and this pile of junk we’ve been dealing with in this thread. Sure the mind can imagine all kinds of things, but that does not make it so.

You have a point, e.g. at one level we can verbalise, “square circle,” and then we may analyse further and see, not a possible being. That comes about because the core characteristics stand in mutual contradiction. Not even God could make a square circle.

But in turn, that brings to bear how inherently mental conceptions and contemplations can have powerful impact in the real world. Square circles etc are forbidden beings, but also — as I argued in 351 — natural numbers are necessary beings, automatically present in any possible world:

Logic — as abstract a thing as we get — constrains reality, and so holds reality in some form. (Which is a big clue. And no, I am not playing at platonism, I am highlighting that the evidence we have points to the fundamental power of ideas in reality, and as ideas seem inextricable from minds, to mind as a foundational aspect of reality. Mathematics, the Achilles’ heel of scientific-technical materialism.)

But, we can start with things such as how natural numbers are necessary beings, and that in a transfinite succession. More or less following a path trod by von Neumann (almost, as usual):

{} –> 0
{0} –> 1
{0, 1} –> 2
{0, 1, 2} –> 3
. . .

OMEGA . . .

EPSILON-NOUGHT

. . .

Strictly, ordinals so far, to get to cardinals, toss away the successor pattern, and for the nth in the sequence from 0, the cardinality of the number is effectively n – 1. That allows us to identify Aleph-null as holding cardinality of the set of natural numbers.

Note those pesky ellipses, we are pointing to an in-principle, a supertask we cannot actually complete, but can contemplate logically — hint, hint on the contemplative, rationally envisioning mind. And, with a suitable set-builder procedure, we can appreciate that the natural numbers cannot not exist, appearing as a direct consequence of a successor process applied to the empty set and a cardinality assignment operation.

Individual naturals are necessary and the set, which is patently transfinite, is also necessary, it cannot not exist in any possible world. Basic Arithmetic operations follow, per logic and things like 2 + 3 = 5 are necessarily true and in simple cases are self evident even to finite, fallible creatures such as we are . . .

From this, I continued to show how we may construct the real number line by recognising (using decimals for convenience, many bases are possible) that W.B, whole + fraction, extends this powerfully:

We can take the number line as a useful construction, and look at the interval, [0, 1). We may define a proper fraction as a ratio of two natural numbers p:q, with q > a, interpolating in the interval. We may then transfer to the place value notation system, and might as well use base 10. Thus we see rationals as WHOLE + FRACTION, (partly) filling the gaps between successive numbers, 0, 1, 2, . . .

If we extend by allowing a series, and the usual notation, whole being W and fraction = B, W + B –> W.B, i.e.:

W + b1/10 + b2/100 + b3/1000 + . . .

With W = . . . w3 X 1,000 + w2 x 100 + w1 x 1, any number on the line can be expressed

. . . w3w2w1.b1b2b3 . . .

where, the ellipses indicate infinite series. Countably infinite, i.e. we apply as many digits as are in a set of cardinality Aleph-null. (In many cases we will have a huge array of leading and trailing zeroes. Which, from grade school on we usually ignore.)

It can be argued that for any given W.B1 and W.B2, we may interpolate another number, for convenience, the average of the two, or some intermediate at any rate. Thus, again in principle, we fill in the spaces, so that gaps between the rational numbers are covered. That’s where irrationals and transcendentals lurk, and once we identify any given one, we can arguably show that each is a root of at least a countably infinite set of others, by applying multiples and fractions etc. pi, 2*pi, 3*pi etc and pi/2, pi/3 etc.

So, we have a case of our finitude and pointing to what lies beyond finitude. The numbers we can actually directly handle and compute out to actual expression form a fine dust on the real number line, but the line is continuous per the concept of interpolation. And, extends without limit.

So, we may then go for negatives by the concept of additive inverses: (- r) + (+ r) –> 0, which is the identity element for addition, r + 0 –> r. Onwards, introduce sqrt (-1) –> i which allows every quadratic to gave a solution, and we see that i*i*r –> -r, and by extension i*r is orthogonal to the reals line. This gives us a 2-d planar space, with z = a + i*b as general co-ord of a complex number. In effect we have vectors that bring in rotations etc and can be handled algebraically with astonishing powers. we have reals as x axis, imaginaries as y axis, and space pops up. Extend to ijk unit vectors and a 3-d algebraically accessible mathematical world is there. Set trajectories based on x(t), y(t) for time and ideas of particles, inertia, momentum, energy etc and physics walks in. Indeed this is akin to how computer sims can be done (with of course discretisation and rounding.)

Is that mental world of forms a non-being?

Not if it has power to speak into the instantiated world as we see around us.

Are things such as the continuum or even the natural numbers, with all the infinities associated, nonsense and irrelevance?

Nope, they are very relevant and they are a big hint. They point to our finitude and limits, and they point beyond finitude. We can only imagine, point to a trend and put in the triple dot ellipsis. But, we see the real world power of mental ideas that point beyond us.

Indeed, we see necessary beings.

Eternal, uncaused realities that are inherently mental.

Blend in that if nothing — utter non-being — ever was, such would forever obtain, as non-being has no causal powers. Multiply by that we just saw infinities of inherently mental things with power that shapes what is possible.

We have a mind-shaped shadow cast on physical reality.

An eternal, mind-shaped shadow, as necessity plus actual reality implies that something always was.

Eternal, creative mind, as candidate no 1 to cast that shadow.

We are looking at the shadow of God.

Again.

We live in a God-haunted world, that reminds us of its roots at every turn.

So, perhaps, just perhaps, these things point somewhere.

As for issues on transfinite cardinalities, we need to take the triple dots seriously. The process, taken stepwise in order, never terminates. Instead, we must see such as laying out the in-principle order that would sequentially give us a set. The set, identified, can then be understood as actually forever present in full, hence its scale or cardinality is transfinite.

And, by applying operations that are logically consistent to transfinite sets we can see transfinitely scaled cardinalities. Where, two sets of entities have the same cardinality if they can be matched, element by element.

Five fingers with five, etc.

At the bar of the transfinite, where we move beyond what we can fully effect (WIKI: >>numbers that are “infinite” in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite . . . >>, where >>The Absolute Infinite is mathematician Georg Cantor’s concept of an “infinity” that transcends the transfinite numbers . . .>> i.e. there are distinctions) but can indicate, we see astonishing properties reminiscent of quantum strangeness:

1, 2, 3, 4 . . .

x 2:

2, 4, 6, 8 . . .

– 1:

1, 3, 5, 7 . . .

That is, understanding cardinality as correspondence and reckoning with the triple dot effect, we see that because the sets are transfinite, the full set of naturals may be matched with proper subsets, the evens and the odds. Indeed, this is the property of inexhaustibility plus set-matching that Cantor used to designate transfinite cardinality. Here, aleph-null. (I suggest here for an introductory read: http://www.ias.ac.in/resonance.....8-0068.pdf and here also: http://www.cogsci.ucsd.edu/~nu.....rgmtcs.pdf )

Is this a case of blatant self-contradiction?

Not if you and I reckon with the impact of the triple dots, this keeps on endlessly, and we can accept the full sets. From this, we may move to even higher order transfinites, which are beyond countability. Most notoriously, the continuum.

Enter, stage right, the hyper reals and their inverses, the all but zero infinitesimals.

And, with them, points as locations without extension. As, implicitly, we have been using all along. Thence too, non-standard analysis and Calculus. This last, the crown jewel of and magic key to the scientific revolution. A mathematical achievement.

Such then become abstract models, forms if we will, that cast a long, long shadow on the physical world.

Whatever else he is, God is a mathematician of the first order.

And so also, utterly logical.

BTW, Cantor strongly associated the Absolute Infinite with God.

But also, out there somewhere, Plato’s ghost is laughing.

And he reminds us, from The Laws, Bk X:

Ath. Then, by Heaven, we have discovered the source of this vain opinion of all those physical investigators; and I would have you examine their arguments with the utmost care, for their impiety is a very serious matter; they not only make a bad and mistaken use of argument, but they lead away the minds of others: that is my opinion of them.

Cle. You are right; but I should like to know how this happens.

Ath. I fear that the argument may seem singular.

Cle. Do not hesitate, Stranger; I see that you are afraid of such a discussion carrying you beyond the limits of legislation. But if there be no other way of showing our agreement in the belief that there are Gods, of whom the law is said now to approve, let us take this way, my good sir.

Ath. Then I suppose that I must repeat the singular argument of those who manufacture the soul according to their own impious notions; they affirm that which is the first cause of the generation and destruction of all things, to be not first, but last, and that which is last to be first, and hence they have fallen into error about the true nature of the Gods.

Cle. Still I do not understand you.

Ath. Nearly all of them, my friends, seem to be ignorant of the nature and power of the soul [[ = psuche], especially in what relates to her origin: they do not know that she is among the first of things, and before all bodies, and is the chief author of their changes and transpositions. And if this is true, and if the soul is older than the body, must not the things which are of the soul’s kindred be of necessity prior to those which appertain to the body?

Cle. Certainly.

Ath. Then thought and attention and mind and art and law will be prior to that which is hard and soft and heavy and light; and the great and primitive works and actions will be works of art; they will be the first, and after them will come nature and works of nature, which however is a wrong term for men to apply to them; these will follow, and will be under the government of art and mind.

Cle. But why is the word “nature” wrong?

Ath. Because those who use the term mean to say that nature is the first creative power; but if the soul turn out to be the primeval element, and not fire or air, then in the truest sense and beyond other things the soul may be said to exist by nature; and this would be true if you proved that the soul is older than the body, but not otherwise.

[[ . . . .]

Ath. . . . when one thing changes another, and that another, of such will there be any primary changing element? How can a thing which is moved by another ever be the beginning of change? Impossible. But when the self-moved changes other, and that again other, and thus thousands upon tens of thousands of bodies are set in motion, must not the beginning of all this motion be the change of the self-moving principle? . . . . self-motion being the origin of all motions, and the first which arises among things at rest as well as among things in motion, is the eldest and mightiest principle of change, and that which is changed by another and yet moves other is second.

[[ . . . .]

Ath. If we were to see this power existing in any earthy, watery, or fiery substance, simple or compound-how should we describe it?

Cle. You mean to ask whether we should call such a self-moving power life?

Ath. I do.

Cle. Certainly we should.

Ath. And when we see soul in anything, must we not do the same-must we not admit that this is life?

[[ . . . . ]

Cle. You mean to say that the essence which is defined as the self-moved is the same with that which has the name soul?

Ath. Yes; and if this is true, do we still maintain that there is anything wanting in the proof that the soul is the first origin and moving power of all that is, or has become, or will be, and their contraries, when she has been clearly shown to be the source of change and motion in all things?

Cle. Certainly not; the soul as being the source of motion, has been most satisfactorily shown to be the oldest of all things.

Ath. And is not that motion which is produced in another, by reason of another, but never has any self-moving power at all, being in truth the change of an inanimate body, to be reckoned second, or by any lower number which you may prefer?

Cle. Exactly.

Ath. Then we are right, and speak the most perfect and absolute truth, when we say that the soul is prior to the body, and that the body is second and comes afterwards, and is born to obey the soul, which is the ruler?

[[ . . . . ]

Ath. If, my friend, we say that the whole path and movement of heaven, and of all that is therein, is by nature akin to the movement and revolution and calculation of mind, and proceeds by kindred laws, then, as is plain, we must say that the best soul takes care of the world and guides it along the good path. [[Plato here explicitly sets up an inference to design (by a good soul) from the intelligible order of the cosmos.]

In short, Plato answers the evolutionary materialist case by making a cosmological inference to design, including in his argument the human experience of being a conscious, minded en-souled creature. On that premise he intends to found public morality, thence an objective basis for just law. Which is his main target here.

Such a conclusion and policy programme, we may indeed choose to reject, but we cannot justly ignore nor censor it out of our considerations by a priori imposition of materialism dressed up in a lab coat.

So, then, I would not be quick to dismiss the issues linked to the mathematics of zero, infinity and their near-neighbours, so to speak.

Hey, mon, let’s enjoy the twelve days of Christmas and the new year as it comes!

The exact same logic can be used to compare the set of natural numbers with the set of real numbers, how can you not see that?

Yeah, you can compare the reals to the natural numbers. And you will find that there are more real numbers. The cardinality of that set is bigger. There is no way to put them into a 1-to-1 correspondence. Look it up if you don’t believe me.

What is the first number in the set of real numbers? I will choose to match the number 1 with that. What is your second number? Since you can’t even say what the second number is, the problem isn’t in the number of elements in any one set, the problem is in you defining what elements are in your set, and what elements aren’t. As soon as you can tell me two of the elements, or three of the elements, then I can tell you what the matching number is from my set of natural elements-its as simple as that.

There is nothing that says I have to match up a one in my set with a one in your set. I can just as well match the number 1 in my set, with the number for Pi in your set. Ok, next….

You would have to be a crazy person (Cantor) to not see how easy this idea is.

No matter what list you made of the real numbers it’s possible to prove you missed one. It’s possible to find one that isn’t matched with a natural number. Since you can find one that isn’t matched the set of reals must be bigger..

You can look up the proof. It’s quite straight forward. And was done over 100 years ago. This is NOT controversial except on UD. This is not materialist dogma or related to ‘Darwinism’. It’s basic set theory. Get Irving Kaplansky’s book, there must be 1000s of used copies about. Or get it through inter-library loan. Actually any set theory book will do.

The set of real numbers cannot be matched up 1 for 1 with the counting or natural numbers therefore the sets are different sizes.

Ph, pardon, there is no first real, the set runs all along the number line on both sides of 0 without upper or lower limit. Even if we specify magnitudes only |r| then for any number beyond zero we please 0.B, between 0.0 and + 0.B we can always specify another one by some means of interpolation. Another astonishing property, infinite fineness, if you will; thus also inherent non-countability which is not a contradiction. That is a direct implication of continuity. And, continuity makes sense coming from other directions, a lot of sense indeed, hence algebra, calculus, analysis etc. Specify, y = 2*x^2 + 1, where x is any real number. This is abstract, a parabola, and the graph we may sketch is yes a representation but that does not mean that the mental construct has no reality or physical influence — we can use this to construct a mirror for a search-light knowing that it will convert the light from a bulb at the focus to a parallel beam (car headlamps and flashlights apply this), or use it in a telescope and be confident that it will focus light from the stars . . . as Newton did in inventing the reflecting telescope. And, along the curve of that parabola, it makes sense to speak of the slope of the curve, thence differentiation, and of the area under it too, or the volume enfolded if we revolve around the y axis, hence wine glasses. And more, ever so much more. KF

Jerad, actually, no. Infinity is ALWAYS controversial. Abstract inherently mental realities tied directly to infinities and their near neighbours are inevitably strange, challenging and open to controversy. What is happening here, is that people are expressing their inner questions, doubts and frankly suspicions regarding an elite academic culture that has repeatedly cynically manipulated and betrayed their trust. For case study no 1, try EUGENICS, that extension of Darwinism, and linked Social Darwinism, which cost dozens of millions of lives within the past century or so. They have a right to be concerned, even to be suspicious. There is a price to be paid when a profession repeatedly violates the ethics of duties of care to truth and to limitations of knowledge claims, and you are seeing here at UD the tip of an iceberg people like us have warned about. KF

I have no idea who this post is in reference too. I think at times your answers are a bit too obtuse to really lead anywhere.

Jerad,

If you say that the set of real numbers is larger than the set of natural numbers, then first you must give some examples of your set. I will prefer to use 1 million as the first number in my list, now what is yours?

And really, if you are going to argue a logical point, please stop trying to refer to some vague authority of, look it up online; online is not an authority on anything, any more than Cantor is. if his argument can’t hold up to logic, its not a valid argument.

Now, my set starts with 1 million and proceeds in increments of 72, what is the first element in your set of real numbers?

I ask this because does not Cantors proof rely on the set of natural numbers being placed in some order. If that requirement is demanded to be offered in one set, then to be consistent it must be presented in the other.

The only way Cantors argument makes any sense at all is if we allow him to make the rules, and insist that the natural numbers follow a predetermined order, but the real numbers have no such duty. Why must one follow Cantors rules?

Can you answer that with words and not just symbols?

I’m not going to try and list the real numbers because i know it can’t be done ’cause no matter what list you give me i can always find one you missed.

I’m not suggesting ‘online’ is an authority. I am suggesting you can find pertinent materials, resources, information, discussions and examples online. I am saying you don’t have to believe me, you can find answers and other people’s work if you really are interested.

If you just want to argue with me then please learn the math you want to argue about. Find out what’s already been worked out.

Ph, The point is, in part that we cannot identify a first — or, perhaps, better, a second — real. We can see 0 as first real, in a certain sense, but then any specific non-zero real, however close to zero, can be used to construct another value intermediate between it and zero. Above, I used the idea of place-value notation decimal numbers of form 19.78 etc which are WHOLE + FRACTION, I used W.B. In mathematical terms the non-zero reals are open intervals (0, infinity), ( negative infinity, 0) on the real number line. The round brackets mean, that there is no terminating value. Any specific value + 0.B or – 0.B will be such that no matter how close to zero, we can interpolate a value closer to 0. But, the details lie in university level Math. Sorry about that, I am not pulling rank, I am simply describing the level at which such is typically studied in technical terms. KF

PS: Symbols are a technical language resorted to because they give enhanced precision. In fact, I am here fighting lack of access to graphical and non-Latin symbols in comments. Sorry, the apparatus is technical. However, in the above I referred to decimal numbers, the real number line and the like, that are accessible. I symbolise a general decimal number W.B for Whole + fraction, and assume understanding of the place value weighting.

This is the whole point. You know you can’t show where I am wrong. You can’t list the first two elements of your set of real numbers. Until you can list them, how can you know that I can’t find a correspondence in the natural numbers?

You are trying to dodge the question, by saying it has already been answered, by someone, somewhere. Its typical materialist, scientific skeptic dodging. Its is very recognizable to anyone who questions Darwinists.

Ph, are you aware, that the glyphs used to represent words by chaining them are symbols with an associated calculable information content? That something is represented based on symbols does not count against its being meaningful or reasonable. I have pointed to primary school level mathematical concepts, which I think we can safely use, decimal numbers, real number line. Then, I pointed out that if you move away from zero to any specific number W.B (e.g. 19.78, or 0.00001978 for example) it will be such that you can ALWAYS construct another number closer to zero. One easy way is, take the average, (p + q)/2. Here, let p = 0, and you will see that the average will be at the mid-point of the line of numbers connecting 0 and q. For 19.78, that mid point number is 9.89. For the second number given, it is 0.00000989, per simple calculation. So, the fact of there being no “second” real, away from zero, points to how such is transfinite, and in fact to how there is an uncountable, much larger number of real numbers than of natural numbers such as 0, 1, 2 . . . and yes, the triple dots for something left off point onwards. These things may be unfamiliar, and may seem strange but they were just next door to what we all did in elementary education. Just, the structure of such education, very wisely, deferred the mysteries until later on, as most people have no practical need to address them. But if you deal routinely with analogue vs digital electronic systems, you have need. My simple illustration in a first digital electronics class was that you must stand on the rungs of a ladder but can hang on to a rope anywhere along its length. Extend that to the ideal case and the rest follows. And if you think this one is hard to swallow, try out quantum mechanics — which happens to be the best empirically supported physical theory, never mind the puzzles, paradoxes and more. KF

PS: Just to pick up one of Zeno’s paradoxical points, if the man running after the tortoise in a first interval covers half the gap, then in the second, half the intervening and so forth, he can and does overtake as common-sense observation shows. For each half-gap step takes a shorter and shorter time and within a finite time attains the limit of passing the tortoise. And so forth.

PPS, the issue is not a fair comparison, but to achieve an accurate and reasonable view. If it is so that we can demonstrate that there is no first non-zero real number, then we should accept that.

You will not be able to understand Zenos paradox nor the problems with cantors argument if you insist that it is only a mathematical problem and not a logical one.

I understand that math is your field, so it is a disadvantage to you, to make you use thinking outside of your math training to answer a question. However, I suggest to you, that in order to debate someone, you will need to step outside of your comfort zone.

If the only way to accept Cantors argument is to apply rules to one set, that you don’t apply to the other set, you have not made a comparison. You are simply wrong.

The only way to apply Cantors logic is to distort what it means to compare. Thus the statement that one set contains more than the other is a non-starter until you realize that it is a logic problem, not a math one.

This is the whole point. You know you can’t show where I am wrong. You can’t list the first two elements of your set of real numbers. Until you can list them, how can you know that I can’t find a correspondence in the natural numbers?

That was your list of the real numbers? Add 72 each time?

You’ve missed out a lot of real numbers there dude.

Why should I make a list, you’re the one making the claim that runs counter to established results. Seems to me you have to prove your point.

If you think you can find a 1-to-1 correspondence between the reals and the naturals numbers by all means have a go. I don’t think you can so why do I need to come up with a list? I know it’s not possible to do so. Any list I make I can prove is wrong so why would I bother?

You think you can then go ahead.

I’d start with zero if I were trying. Then I’d try to list the infinity of numbers between zero and one. I’d try and find a scheme where nothing was left out. See if you can make that work.

Of course the problem is anytime you tell me there are two reals with nothing I between them I’d just find one. The gaps get smaller and smaller indefinitely.

You do know what the real numbers are? You do know that sqrt(2) is a real number. And pi. And e. Stuff like that.

If the only way to accept Cantors argument is to apply rules to one set, that you don’t apply to the other set, you have not made a comparison. You are simply wrong.

The only way to apply Cantors logic is to distort what it means to compare. Thus the statement that one set contains more than the other is a non-starter until you realize that it is a logic problem, not a math one.

Where did you read Cantor’s proof ? You seem to have a lot of wrong notions about Cantor’s proof. Here are the basics (includes Cantor’s proof) in simple form. You can see that Cantor’s proof is logical. In fact since infinite sets can’t be written out, the only way to show it is not countable is by logic ! It doesn’t have any heavy maths at all.

Try to think, don’t just let others do it for you.

What are the rules for comparison between the set of real numbers and the set of natural numbers, are the rules applied equally? If they are not, then my set is bigger, it contains fairies.

What Cantor has done is a simple card trick. Anyone can play it. The object is to name a card that is the same as the other players card. Whoever can name the most cards the same as the other player wins.

But there is one catch, you have to always name your card first, then I name mine second. I will always win.

The question is, why does Cantor decides who lays down their card first?

How about we alternate who goes first, so that it is a real comparison, ok, I will start, 11, what is your match?

Now when we play a fair game of comparison, no one wins. It will always be a draw. No matter what you name, I can match it, and no matter what I name you can match. Its a draw! Pretty darn simple for anyone who can think without a calculator.

Does my set of natural numbers have to be in order, or not? If they do, then so do yours.

They do not have to be in order as long as you can keep track of them.

if they don’t, then let’s keep it simple, what corresponds with the last of my numbers in the list of natural numbers?

There is no last number in your list. If you’re talking about starting with a million and keep adding 72 indefinitely

If you want to say that there need not be rules applied equally to both sets, then my set is bigger, because it contains fairies, that yours does not.

I expect you to use terms in the same way as mathematicians do or give a good solid definition of the terms you do use so I know specifically what you mean.

Anyway, you seem to be missing the point: you think you can demonstrate a 1-to-1 correspondence between the reals and the natural numbers. You haven’t yet shown me your matching scheme between the two sets. Or given me hint of what it might be. You’re talking about a set of integers for some reason. If you’re going to have a list of reals then sqrt(2) had better be on it.

If your going to make an argument that’s fine but you haven’t yet. This isn’t rocket surgery but it’s not a parlour game either. If you’re not going to take it seriously then don’t bother..

I can match any element in your set with an element in my set, if you think I can’t just tell me one element in your set, I can’t match from mine? Its that simple.

I can match any element in your set with an element in my set, if you think I can’t just tell me one element in your set, I can’t match from mine? Its that simple.

I’m sure you could. But that’s just showing a 1-to-1 correspondence between things I pick and your set. I believe you can do that. But that doesn’t show a 1-to-1 correspondence between the reals and the natural numbers.

What if my first number was the cube root of the imaginary number i? What if my first number was e^(-3i)? What if my first number was 428 mod 7? What if my first number was aleph-one? What if my first number was 0/0? Are you just going to say:add 72 to it, that’s my number?

That’s not showing a 1-to-1 correspondence between the reals and the natural numbers since I can’t list all the reals (as I said) which means we’d miss a real number just as I said.

What would e^(-3i) + 72 be anyway? Do you know? Is it in your set? Is sqrt(2) + 72 in your set?

No Jerad, you seem confused. I said I can match any element in your set with an element in my set, and still have some leftover. Its no different than what Cantor claims. Cantor simply plays a card game by saying the naturals must reveal their order first, then allow him to counter with his owned preferred one to one match.

If I turn it around and say the reals must reveal their set first, and I get to match up one to one, Then I win the game.

Ph, logic is not only a branch of Phil, it is also a branch of Mathematics. And, the point of an open interval is that there is no specifiable first element. I have already shown that once you move away from zero, you have a case that for any given element, we can specify another closer to zero. Which of course feeds into the Zeno type issue, and through the physics of a series in time, we see why there is in fact a logical-mathematical basis for the easily observed fact that Achilles overtakes the tortoise in a calculable interval. In effect the additional increments of time and associated space converge to a definite finite time and space point (as each successive increasingly small increment of space takes a correspondingly small successive increment of time), the place and time of overlap, and after that A is ahead of the tortoise. As is familiar from simple observation. KF

PS: Applied Physicist actually, though I did pick up a third major, in math, in my u/grad work.

PPS: For the naturals, start with ordinals, and I already outlined the process that naturally yields the set as a set-builder process. To move to naturals, roughly speaking toss away the sequence ordering ops, to define cardinality. Thence, to get to the rest define operations etc as done in primary school, and get to the decimal numbers. That puts us in the space of reals once we see that we have whole number part and fractional part and can extend through addition so we have the place value notation representation of what is in effect an infinite series. Once there, we see continuity emerging and as a consequence, there will be no first real different from zero, we have an open interval. As was already pointed out in outline, I am just summarising. And, symbols allow us to compress the process, which in words is much like the descriptions of the Temple and Tabernacle in the Bible. Unwieldy and confusing because the flood of words gets us lost in the details.

Then conversely mathematics is a branch of philosophy, so the philosophy must first be correct, for the mathematics to work.

If there is no first element in your set, then there is no first element in mine. We are at a stalemate. Cantor loses.

Are you sure you understand Zenos paradox of Achilles? The paradox is that he never passes the tortoise, if we stack the rules so that he can only cover half the distance each time. Likewise, Cantor’s “paradox” only works if you stack the rules.

I have, you ignored it and prattled on as if you haven’t been refuted.

Two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set. And that also means there isn’t any real 1-to-1 correspondence.

Deal with that or you will prove that you cannot deal with reality.

With your example I have found an there are an infinite number of unmarried elements and using the mathematics of sets we have found them. That is by subtracting all the elements of one set from the other we, together, have demonstrated there are infinite elements left in one set but none the other. That means there are infinite elements left unmarried.

And using the same mathematics of sets I have shown that your formula actually shows the difference in size between two sets with infinite elements. I know that upsets you but too bad.

Everyone sees that you are afraid to deal with that, Jerad. Thank you.

Are you sure you understand Zenos paradox of Achilles? The paradox is that he never passes the tortoise, if we stack the rules so that he can only cover half the distance each time. Likewise, Cantor’s “paradox” only works if you stack the rules

Zeno’s paradox is akin to geometric series: sum of (1/2)^1 +….(1/2)^inf = 1, the sum converges, hence the distance can be covered by Achilles. There is no ‘stacking the rules’.

There is nothing wrong with Pi. One should calculate Pi to whatever precision is required by the problem one wants to solve. Pi is the perfect example of the stupidity of continuous structures. Nobody can fully calculate Pi simply because continuity is nonsense. In nature, there is a limit to how smooth a circle (or any curvature) can be and this limit is enforced by the Planck length, i.e., the smallest fundamental distance between two particles.

Maybe we’re not as far apart as I thought, then. Here is what I surmise from your post:

1) Pi exists as a real number.

2) You can estimate pi to any degree of accuracy you wish (presumably using a series representation?)

3) In fact, once you’ve chosen a series representation, if you choose a small number ‘tolerance’, there is a natural number k such that all the partial sums s_n with n > k are within ‘tolerance’ of pi. In other words, the sequence of partial sums converges to pi.

But then, by definition, we could write pi equals the original series representation.

I wonder if that is still acceptable to you, as earlier in the thread you stated that no infinite series exist.

MT, of course, the trick is the time steps drop with the distance steps and the infinite sequence descends to the infinitesimal, leading to the limit at the overtake point. Zeno “missed” that part. KF

phoodoo: Does my set of natural numbers have to be in order, or not?

No, but you have to show that you have accounted for every element of the set. So if we take the rational numbers, we can show a one-to-one correspondence with the natural numbers. If we take the real numbers, we can show that no such one-to-one correspondence is possible.

Infinity doesn’t work the same way as finite numbers. A good example is Hilbert’s veridical paradox of the Grand Hotel.

Consider a hotel with an infinite number of rooms and all the rooms are filled. A new guest arrives. We can move the guest in room 1 to room 2, the guest in room 2 to room 3, the guest in room N to room N+1, and so on. Now room 1 is open for the new guest. We can do this again and again. The Grand Hotel is always full, yet always has room for more.

DS, yup, once we move from the abstract domain of mathematical forms to calculation, we move to discrete approximate but useful values applicable to the physical world. As was noted hundreds of comments ago. Thus, we duly note the differences and proceed. KF

PS: in so proceeding, let us note that decimal, place value notation numbers such as 1.978 or 197.8 or 0.000001978 etc are all handy sugar-coats for INFINITE POWER SERIES on powers of ten, in form W.B. Such are in principle — beyond some partial sum and a relevant delta neighbourhood, going to be within that delta of the targetted real value, convergence guaranteed. Yup, infinite series sitting next door to grade school math. And with digital signal processing and Z transforms begging to come in the door also. (With the neighbouring Laplace Transforms in the briefcase.)

Ph, yup, EVERY intellectual domain targetting knowledge fits in with philosophy, the love of wisdom with particular focus on hard questions. Newton’s major work was in effect Mathematical Principles of Natural Philosophy. And the Ph.D degree boils down to, teacher of the love of wisdom. What we do is to hive off a new discipline when it seems reasonable to assign it a specialisation. But the meta-issues that come in with phil and issues such as logic, epistemology, metaphysics, ethics and so forth never actually go away. They tend to resurface when a paradigm is running into trouble, especially, and that causes a lot of trouble for those unaware of the issues that come with them or the required special approach of comparative difficulties. Hence BTW the significance of Phil-of-X meta-disciplines, such as for X = Science, Math, History, Religion, etc. KF

PS: The logic involved in the man chasing the tortoise, is that we have a double-acting series, in which both spatial and temporal steps are converging on a limit. As you get more and more shorter steps, you do them just as much faster and faster, so the tortoise gets overtaken at a predictable finite point in space and time. The calculus-algebraic expression will be abstract, and the series expression will converge on the same value. Reading (HT BA77):

Maybe we’re not as far apart as I thought, then. Here is what I surmise from your post:

1) Pi exists as a real number.

Certainly not. 3.14 is a number. Pi is a symbol that represents a principle, function or calculation. I have always felt that Pi should be written with a different notation to indicate how many digits one wants after the decimal point. For examples, Pi-2 or Pi-16, etc.

2) You can estimate pi to any degree of accuracy you wish (presumably using a series representation?)

3) In fact, once you’ve chosen a series representation, if you choose a small number ‘tolerance’, there is a natural number k such that all the partial sums s_n with n > k are within ‘tolerance’ of pi. In other words, the sequence of partial sums converges to pi.

But then, by definition, we could write pi equals the original series representation.

The key concept here is this: “if you choose a small number ‘tolerance’”. And it is not a matter of if, either. One has to choose, otherwise nothing can be computed.

I wonder if that is still acceptable to you, as earlier in the thread you stated that no infinite series exist.

Not really. I repeat. There is no such thing as a truly smooth circle, a million fruitcakes claiming otherwise notwithstanding.

Infinity doesn’t work the same way as finite numbers. A good example is Hilbert’s veridical paradox of the Grand Hotel.

Consider a hotel with an infinite number of rooms and all the rooms are filled. A new guest arrives. We can move the guest in room 1 to room 2, the guest in room 2 to room 3, the guest in room N to room N+1, and so on. Now room 1 is open for the new guest. We can do this again and again. The Grand Hotel is always full, yet always has room for more.

By the way, how many natural numbers are there?

Zacky-O is the biggest fruitcake on this thread and probably every other thread as well.

and in fact to how there is an uncountable, much larger number of real numbers than of natural numbers such as 0, 1, 2 . . . and yes, the triple dots for something left off point onwards.

(1) there are just as many even numbers as natural numbers!

(2) there are just as many natural numbers as rational numbers!

(3) there are more irrational numbers than rational numbers!

(4) there are more subsets of natural numbers than natural numbers!

Certainly not. 3.14 is a number. Pi is a symbol that represents a principle, function or calculation. I have always felt that Pi should be written with a different notation to indicate how many digits one wants after the decimal point. For examples, Pi-2 or Pi-16, etc.

I guess some differences still remain.

And I can see the logic behind this notation, but what about cases such as the formula A = pi * r^2? Doubtless no exact circles physically exist, but we often do need to estimate the area of an approximately circular region. Or even find the radius of such a region given its area. I’ve always just used the symbol pi on its own, and have never reached the wrong answer to an applied problem by treating pi as a genuine real number.

Doesn’t that indicate that treating pi as a real number is, at the very worst, just a convenient shortcut? And that perhaps we should reconsider what we mean by “number”, possibly including things such as pi which can be expressed as the limit of a convergent sequence of rationals. Seems to me to be a very pragmatic step.

The real numbers have the property that they are ordered, which means that given any two different numbers we can always say that one is greater or less than the other. A more formal way of saying this is:

For any two real numbers a and b, one and only one of the following three statements is true:

1. a is less than b
2. a is equal to b
3. a is greater than b

Philosophy is very good subject for killing time – make up something and talk about it as if it is profound. The philosophy of philosophy is to make sure that whatever is made out to be philosophical should be of no practical use.

Me_Think now:

You can see that Cantor’s proof is logical. In fact since infinite sets can’t be written out, the only way to show it is not countable is by logic ! It doesn’t have any heavy maths at all.

Doesn’t that indicate that treating pi as a real number is, at the very worst, just a convenient shortcut? And that perhaps we should reconsider what we mean by “number”, possibly including things such as pi which can be expressed as the limit of a convergent sequence of rationals. Seems to me to be a very pragmatic step.

I have no problem with this, provided that one realizes that there is no such thing as a true circle. There never was and never will be. Sooner or later, Pi must be substituted with an actual value to obtain some result. I use Pi all the time in my speech recognition research. But my computer is finite just like everything else and I have to use an actual floating point number that is good enough for my calculations.

The natural numbers N: {0,1,2,3…}
Those numbers squared ‘N^2’ :{0,1,4,9…}

And I take it your argument is that that because there is a subset of N not present in N^2 then N must have a cardinality Card(N^2) + Card(N \ N^2) which is > Card(N^2) as long as Card(N \ N^2) > 0?

Here’s the problem: the cardinality of N, N^2 and N-N^2 are all the same, and so is the their sum.

Any infinite subset of a the natural numbers has the same cardinality as the naturals. We can show this by creating a “general” one-to-one correspondence:

For any subset “S” of the natural numbers, there will necessarily be a minimum value. We can map that value the lowest natural number. If we call the mapping function `f` then f(0) = s_0. The remaining elements in S must also have a lowest value, so we can keep going, mapping the lowest value in our subset to the next natural number:

f(1) = s_1, f(2) = s_2 …. f(k) = s_k.

That obviously maps a natural number to a member of the subset. The same relationship also uniquely maps any member of the subset to a natural number: for any k there are only a finite number of values lower than s_k (there are k), so, the third non-square number maps to ‘3’ in the naturals and the k-th number uniquely corresponds to k.

So, for any subset of the naturals, there is always a unique way to map a value in N to one and only one value in that subset. That is, they have the same cardinality, which is called aleph_0 or “countable infinity”.

Now, what does that do to your argument. You are saying

Card(N) = Card(N^2) + Card(N \ N^2) > N^2

But, if we put values we see that equation winds up telling us

aleph_0 = aleph_0 + aleph_0 = 2 * aleph_0

So, what you’ve actually shown is that 2*aleph_0 = aleph_0, which is trueand can be shown by the same “well ordering” trick above. In other words, you approach shows us one of the counter-intuitive properties of infinite sets, and helps establish that the cardinality of these specific (and in fact any infinite) subsets of N have the same cardinality as N.

I have no problem with this, provided that one realizes that there is no such thing as a true circle. There never was and never will be. Sooner or later, Pi must be substituted with an actual value to obtain some result. I use Pi all the time in my speech recognition research. But my computer is finite just like everything else and I have to use an actual floating point number that is good enough for my calculations.

wd400- If cardinality pertains to the number of elements in a set, then two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set. And that also means there isn’t any real 1-to-1 correspondence.

The natural numbers N: {0,1,2,3…}
Those numbers squared ‘N^2? :{0,1,4,9…}

And I take it your argument is that that because there is a subset of N not present in N^2 then N must have a cardinality Card(N^2) + Card(N \ N^2) which is > Card(N^2) as long as Card(N \ N^2) > 0?

The relative difference in cardinality between the two sets would be n squared.

So, what you’ve actually shown is that 2*aleph_0 = aleph_0, which is trueand can be shown by the same “well ordering” trick above.

Truly magical. Unfortunately magic doesn’t belong in math.

Mung, 488: Yup, there are a lot more subsets of the naturals than there are naturals. As in the cardinality of the set of subsets of a set of cardinality n is of scale 2^n. That gives us aleph_1 in this case, which is often held to be the continuum number but that is a hypothesis. KF

Joe, you are thinking in terms of sets with a definitive terminus, i.e. finite sets. Here, we deal with the transfinite. KF

PS: It may be helpful to make a loose analogy to order of magnitude. Naturals, evens and odds are of the same order of transfiniteness, but 2^aleph null is of exponentially higher order.

It occurred to me that the reason that Darwinists love this infinity crackpottery so much is that they need it to support their ‘infinite monkeys banging on typewriters’ hypothesis. After all, the only argument (assuming stupidity is an argument) they have to counter the obvious fine tuning of the universe is that there must be an infinite number of parallel universes and that we just happen to be in one that is well suited for life.

Joe, the sort of count you specify is a supertask and cannot complete as a transfinite cannot be traversed stepwise. You have to deliver the sets all at once or else have the sort of speeding up Zeno implied that completes the series in a limited time. The transfinite is tricky, at any given finite time a double-fast count will exceed the base count, but you have not even approached the transfinite yet. That is why one looks at set-builder logic and compares the scale which as shown yields that say 1, 2, 3 . . . x2 gives 2, 4, 6 . . . less 1 gives 1, 3, 5 . . . and THAT gives the same order of scale. To break out beyond that you need to go to exponentials such as the power set which goes to 2^aleph_null. KF

“The introduction of set theory at the end of the nineteenth century persuaded many mathematicians, Bertrand Russell among them, that they had discovered a system by which the natural numbers could be displaced in favor of something more fundamental. The creation of Georg Cantor, set theory is the most remarkable single achievement of nineteenth-century mathematics, so much so that David Hilbert was moved to call it a paradise.”

– David Berlinksi, One, Two, Three: Absolutely Elementary Mathematics

A circle (at the origin) is the set of every point on the plane that satisfies the condition r^2 = x^2 + y^2 in Cartesian coordinates, or simply r = a in polar coordinates.

A circle (at the origin) is the set of every point on the plane that satisfies the condition r^2 = x^2 + y^2 in Cartesian coordinates, or simply r = a in polar coordinates.

Sorry. A perfect circle is not a formula. It’s an idealized construct that can never exist.

Another example of a fruitcake is Stephen Hawking, the crackpot in the wheelchair whose former wife once said that her duty in the marriage was to remind Hawking everyday that he was not God.

Against all logic, Hawking insists that time travel is a possibility in spacetime or “Einstein’s universe”, as he calls it. And yet, anybody with less than two neurons between their ears knows that spacetime is a block universe in which nothing happens. This is something that did not escape the great Karl Popper who wrote about it in “Conjectures and Refutations” and compared Einstein to Parmenides, the Greek philosopher who denied change and motion.

PS. In my opinion, Parmenides (whose work we know by way of Zeno and Aristotle) did not really deny motion/change. He was just trying to point out that there is something fishy about continuity. But I digress.

I have to disagree with you both. There is one ideal circle and it can be perceived by the mind. The ideal circle does not become a multiplicity just because it is apprehended by several minds. The ideal circle in my mind is the same ideal circle that is in your mind.

All you have to do is name the elements in your set, and I can show you a one to one correspondence from mine.

The problem is not in finding the one to one correspondence, it is in you naming the elements of your set. Give them a name, and I can show you a correspondence in mine.

Can you name Two that don’t have a one to one correspondence in mine? Ok, if you can’t name two, then how about One?

Cantors is not a mathematical proof, it is a slight of hand mental card trick. It is because many mathematicians are caught off guard having to defend words, that we end up with many of the poorly solved realities in the world that Mapou has mentioned.

Until you can name One element in your set, that I can’t find a one to one correspondence with from mine, Cantors work is still just that of an amateur magician in history. You have explained no realities in the world.

Sorry for taking so long to catch up. It’s been a long day

No Jerad, you seem confused. I said I can match any element in your set with an element in my set, and still have some leftover. Its no different than what Cantor claims. Cantor simply plays a card game by saying the naturals must reveal their order first, then allow him to counter with his owned preferred one to one match.

Look, you are making a claim that YOU can find a 1–to–1 correspondence between the reals and the natural numbers. It’s up to you to defend that claim.

I am absolutely NOT going to play that game because I think it’s not possible to find such a correspondence.

You can’t claim a victory because you devised a personal version of what you think Cantor was saying.

Read his argument why there are more reals than natural numbers, it’s easy to find online, and then lets talk. I’m interested in particular criticisms you might have of his method. Please be specific.

If I turn it around and say the reals must reveal their set first, and I get to match up one to one, Then I win the game.

But, as I already said, I don’t think it’s possible to list all the real numbers. Either explicitly or implicitly or recursively. I don’t understand you argument to be honest. You’ve devised your own test. arguing against a strawman which no one is defending.

Cantors law only works if he games the rules.

Cantor says: lets say I can create a recursive list of all the real numbers. It’s a list so I can number it. that gives me a 1-to-1 correspondence with the reals. BUT I can create a real number which is NOT on the list. So the list wasn’t complete. Add that number to the list. And then create another number not on the list.

This is the real point. With countably infinite sets you CAN create a correspondence with the natural numbers. Anytime you try and do that with the real numbers you find a real number which is not matched up. Therefore, the set containing all the reals is bigger than the set containing the natural numbers EVEN THOUGH both sets are infinite.

Mapou and CM, the set builder criterion has specified the circle. And as r may take a considerable and even continuous value of ranges, the family is indefinitely dense. KF

Two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set. And that also means there isn’t any real 1-to-1 correspondence.

And yet I have come up with a 1-to-1 correspondence that you have been unable to find a counter example.

In my example of {1, 2, 3, 4 . . . } and {2, 5, 8, 11 . . .} wherein I link an element n in the first set with an element 3n – 1 in the second set, every element of both sets is accounted for and linked to an element of the other set. Unless you can find an unlinked element of either set.

Deal with that or you will prove that you cannot deal with reality.

I have been doing so. I found a matching/mapping that does what you claimed can’t be done. Why don’t you show me where I am wrong?

With your example I have found an there are an infinite number of unmarried elements and using the mathematics of sets we have found them. That is by subtracting all the elements of one set from the other we, together, have demonstrated there are infinite elements left in one set but none the other. That means there are infinite elements left unmarried.

Tell me one of your unmarried elements from my sets {1, 2, 3, 4 . . . } and {2, 5, 8, 11 . . . } using the matching n in set 1 matches to 3n – 1 in set 2. Tell me an element, a number, in either set that does not get matched to a single element of the other set. No hand waving, no bluster. Give me an example.

And using the same mathematics of sets I have shown that your formula actually shows the difference in size between two sets with infinite elements. I know that upsets you but too bad.

And what is that size difference Joe? And how do you get that from the formula that I provided? Be specific.

Everyone sees that you are afraid to deal with that, Jerad. Thank you.

Unlike you, I will not speak for everyone. But I do await your providing the examples you say exist.

WRONG! 3n-1, as I have explained to you already. It’s as if you are unable to comprehend what I post.

I am sorry if you haven’t been clear. What exactly does that formula tell you about the cardinality of the sets involved? Please be specific. And say how you draw your conclusion from the formula WHICH I PROVIDED. Not you.

Another example of a fruitcake is Stephen Hawking, the crackpot in the wheelchair whose former wife once said that her duty in the marriage was to remind Hawking everyday that he was not God.

Someone please tell me that I am not the only participant who finds this offensive.

Until you can name One element in your set, that I can’t find a one to one correspondence with from mine, Cantors work is still just that of an amateur magician in history. You have explained no realities in the world.

This is pretty much what Cantor did. He said: create a list of all the real numbers and I will link them to the natural number simply by numbering your list. But, I can find a real number that is not on your list even though you think you listed them all. If you then add that number to your list I will create another real number not on your amended list.

You can list, projectively, all the natural, counting numbers. If you try and do the same thing with the real numbers a clever observer will always be able to find one you missed.

What’s offensive is being lied to by famous people with access to a bully pulpit. It’s not nice to deceive the public. I, for one, am deeply offended.

If you’ve got a serious and credible argument against what Dr Hawkins is saying then by all means publish it. But do not slander people just because you disagree with them

KF has taken a strong stance against slanderous comments and behaviour. I wonder what he thinks of your statement.

If you’ve got a serious and credible argument against what Dr Hawkins is saying then by all means publish it. But do not slander people just because you disagree with them

KF has taken a strong stance against slanderous comments and behaviour. I wonder what he thinks of your statement.

You are the one who is slandering me for accusing me of slandering Hawking. What I said about Hawking and his time travel crackpottery is the truth and I can prove it. And yes, I have published my opinion of Hawking on my blog. Sue me if you take offence.

Match an element n in set 1 to an element 3n – 1 in set 2. That means that the element 1 in set one gets matched to 2 in set 2. 2 in set 1 gets matched to 5 in set 2. 3 in set one gets matched to 8 in set 2. etc.

This proves that I can match every element of both sets uniquely to an element of the other set. Tell me an element of either set and I can find its match in the other set. And I further say that this can only be true if the sets are the same size. And I concede that if you can find an element of either set which is not matched up with my criteria then I will have been proved wrong.

So, can you find an element of either set that is not matched up under my criteria?

That’s it. Have a go as we say in England. You find an unmatched element under my scheme and I’ll concede.

You are the one who is slandering me for accusing me of slandering Hawking. What I said about Hawking and his time travel crackpottery is the truth and I can prove it. And yes, I have published my opinion of Hawking on my blog. Sue me if you take offence.

I can’t sue on behalf of Dr Hawking. And I do applaud you for having the chutzpah to stand behind your opinion. At least you’re not some one who takes a shot and then runs.

Whether or not what you claim is true really is true; I shall leave to the physicists. I suspect you are wrong. But I will defend to the death your right to say what you think. I think we can agree that that right matters. A lot.

And yet I have come up with a 1-to-1 correspondence that you have been unable to find a counter example.

LoL! Mine was the counter-example to yours and you have not been able to counter that. I will not continue to refute imaginary matches. The only way you will ever prove your case is to show a practical use that could only exist if it were true. Something like A^2 + B^2 = C^2.

Mathematics is a useful concept and it isn’t philosophy. Bad things tend to happen when we get the math wrong.

Will trajectories be affected if we all agreed that the cardinality of countably infinite sets could be different? Would relativity be refuted? Would the climate ease to change?

What calculations would be affected?

That is if we allowed the standard and accepted definitions and rules of set theory to hold universally, as universal laws do (see the math), meaning ALL proper subsets will have fewer elements than their “parent” or super set, what, in the real world, would be affected? What would be the effect of such a thing?

We could then use the “mapping formula” to tell us the relative difference. For example:

Let set A = {1,2,3,4,…}
Let set B = {2,4,6,8,…}

Would be a relative difference of 2n, meaning set A is twice the size of set B.

And one more time- If Jerad’s methodology was correct I would not be able to use standard set mathematics to prove him wrong. And yet I have.

That’s it. Have a go as we say in England. You find an unmatched element under my scheme and I’ll concede.

AGAIN- IT’S YOUR SCHEME THAT IS BEING DEBATED. That means you cannot just keep trotting it out as support for itself. And you cannot justify using your scheme for the reasons provided.

You are the one who is slandering me for accusing me of slandering Hawking. What I said about Hawking and his time travel crackpottery is the truth and I can prove it. And yes, I have published my opinion of Hawking on my blog. Sue me if you take offence.

I can’t sue on behalf of Dr Hawking. And I do applaud you for having the chutzpah to stand behind your opinion. At least you’re not some one who takes a shot and then runs.

90% of what makes good science is guts, IMO. I have no fear; so I have an advantage over others. It’s too bad science has been invaded by cowards, butt kissers, crackpots and liars. Fortunately, it will all come to an end soon enough. Wait for it.

Whether or not what you claim is true really is true; I shall leave to the physicists. I suspect you are wrong.

This is because you are a crackpot just like Stephen Hawking. I gave the proof of the impossibility of motion in time or spacetime in this thread. It’s not rocket science. So I can’t say that it went over your head. I can only conclude that you, too, are a coward.

But I will defend to the death your right to say what you think. I think we can agree that that right matters. A lot.

I would stand my ground even if I did not have that legal right.

Here’s a question for all those who think the idea of an ideal circle is nonsense. What exactly is the set of all (x, y) in the plane such that x^2 + y^2 = 1? Certainly it’s a nonempty set, as (1, 0), (0, -1), (8/17, 15/17), and a bunch more points satisfy the condition x^2 + y^2 = 1.

If you plot all these points in the plane, they seem to lie more or less on a circle of radius 1.

If it’s not an ideal circle of radius 1, what is it?

I think one of the problems is failure to distinguish establishing the logic that builds a set or matches (or fails to match) members of distinct sets in 1:1 correspondence, with a stepwise exhaustion of the members.

The former is achievable for transfinite numbers, the latter is not.

But, if you are willing to go with the force of logic, the former should be enough on pain of good old selective hyperskepticism. Further to this, in dealing with abstract mathematical objects, we are dealing with a mental, abstract world of forms, variables and the like.

That such a world can be shown to have sufficient reality to influence what happens or can happen in the real world — e.g. no square circles can exist — should give us all pause.

When we turn to physical implementations such as forming a wheel or the disk that we intend to cut gears on in light of a mathematical specification, we cannot fully implement the ideal, not least as there will be atoms distributed in metal crystals, and at practical level it is hard to do things to a precision of 1/1000 of an inch, consistently. (Oops, there was a key skip.)

Ask any machinist, or ask the engineers at Abu as to why the Record reels of the 1940’s – 50’s did not have interchangeable parts. But by 1954 or so, the new Ambassadeurs, did. Likewise parts of the old 0.303 SMLE were not interchangeable, e.g. even though the magazines were detachable, they were specific to a particular rifle (and in early models were chained to it, I gather). That’s why loading was by 5-round charger clips. I don’t know if the later Ishapore India Rifles, now chambered in 7.62 NATO, are fully interchangeable, but on general progress of tech, I suspect so. (Yes, the SMLE — India and Australia retained the WW I model — is still a police rifle, and may hold reserve status in India.)

But, surprise, the mathematical world is able to address these issues, and that is how later models were made with sufficient precision to have interchangeable parts. Hence, statistical process control and quality management.

We are back at Wigner’s remark about the astonishing powers of Mathematics in the physical world. And the implication that logico-mathematical constraints and opportunities were built-in from the outset. Pointing in very interesting ontological directions.

Coming back, the key point is, are we willing to go with the force of the logico-mathematical points when they are sufficiently shown?

If so, then if we can show how 1:1 correspondences exist with transfinite sets and subsets through transformations such as:

1, 2, 3 . . .

x2

2, 4, 6 . . .

-1

1, 3, 5 . . .

. . . then we should accept that.As, here, patently, we have transformed the full set into subsets. Even, when we can also see that trying to subtract elements is problematic. This may seem strange, but the force of the logic is, we cannot stepwise traverse and exhaust the sets, which is a situation that goes beyond the world of our common experience.

We are forced to address pure logic, and it gives unpalatable results for the abstract world of mathematical forms. But, without that world, we lose the logical foundations of Calculus and wider Analysis, with devastating impact. For, calculus was and remains the crown jewel and magic key of the scientific revolution.

LoL! Mine was the counter-example to yours and you have not been able to counter that. I will not continue to refute imaginary matches. The only way you will ever prove your case is to show a practical use that could only exist if it were true. Something like A^2 + B^2 = C^2.

I am not denying your matching gives a different result but you can’t say yours is right and mine is wrong without finding something wrong with mine. Can you find an unmatched element in either set in my matching? I don’t think you can since you haven’t come up with one. If all elements of both sets get matched up with elements from the other set then the sets must be the same size. This is what you have to focus on.

Will trajectories be affected if we all agreed that the cardinality of countably infinite sets could be different? Would relativity be refuted? Would the climate ease to change?

Nope, reality doesn’t care about what we think.

What calculations would be affected?

Can you find an unmatched element in my scheme or not?

That is if we allowed the standard and accepted definitions and rules of set theory to hold universally, as universal laws do (see the math), meaning ALL proper subsets will have fewer elements than their “parent” or super set, what, in the real world, would be affected? What would be the effect of such a thing?

If I’m right then it is possible for a proper subset to have the same number of elements as the parent set. Welcome to the infinite world.

Let set A = {1,2,3,4,…}
Let set B = {2,4,6,8,…}

Would be a relative difference of 2n, meaning set A is twice the size of set B.

Except that you can match up the elements 1-to-1 so the sets are the same size.

And one more time- If Jerad’s methodology was correct I would not be able to use standard set mathematics to prove him wrong. And yet I have.

No, you haven’t. You haven’t found a mistake. You’re just asserting what is true for finite sets. It doesn’t hold for infinite sets. Go look up that rule you keep quoting and see if it says for all sets or only for finite sets.

AGAIN- IT’S YOUR SCHEME THAT IS BEING DEBATED. That means you cannot just keep trotting it out as support for itself. And you cannot justify using your scheme for the reasons provided.

Don’t debate it, find a mistake. Find an unmatched element using my scheme. You keep using finite reasoning for an infinite problem.

Is this not getting through?

You haven’t found a mistake in my matching. You haven’t found an unmatched element. Why don’t you admit that Joe? Your finite rule doesn’t work for infinite sets.

We are forced to address pure logic, and it gives unpalatable results for the abstract world of mathematical forms. But, without that world, we lose the logical foundations of Calculus and wider Analysis, with devastating impact. For, calculus was and remains the crown jewel and magic key of the scientific revolution.

Hear, hear!! Which has the interesting duel meaning of “I agree” and “listen, listen”.

LOL. So you counted all the points on a wheel and you found them to be exactly what? Infinite?

Pi is irrational and Real number and is part of Cantor’s set, yet it is used for real objects. The point is, it doesn’t matter if you think Cantor was a nut. There are irrational numbers in Wheel’s manufacturing.A wheel is a real object which is a circle – that’s all that matters.

MT: Pi is worse, it is transcendental, one cannot construct a polynomial eqn with rational coefficients whose root will be pi. To estimate it for practical purposes we are left with successive approximations by infinite length power series analysis, duly truncated on convergence. Or the like. And such brings us to the epsilon-delta analysis on partial sums. As in, beyond a partial sum of n terms, for some value of tolerance delta, the difference between the partial sum and the elusive final value will be some error epsilon that is demonstrably less than delta. But to get there all the stuff on transfinites, continuum, infinitesimals and more lurk. Not to mention that place value notation decimal numerals (or the equivalent for binary, hex etc) all are infinite series expressions in disguise, typically with suppressed leading and trailing zeros of indefinite but large number, in principle transfinite. Grade school math sits next door to the most profound mysteries. KF

MT, a wheel is close enough to ideal for govt work, but it is never quite perfect. There will be approximations boiling down to effectively a very high order n-gon. It’s like how monofilament line does not have a perfectly consistent shape. Perfect circles are ideal, logical-mathematical, not physical world. KF

Mapou and CM, the set builder criterion has specified the circle. And as r may take a considerable and even continuous value of ranges, the family is indefinitely dense. KF

I’m not really arguing with you, just trying to clarify how we use the term, “perfect”. Your set of circles do not exist in the material world nor do they exist in the ideal realm. They are a mathematical construct or tool by which we can approximate to the one ideal circle, actual material circles or circles we have thought up . None of your perfect circles can exist in any absolute sense because they each have magnitude which is relative. The absolute, ideal circe is a unity which exists in and of itself and is not relative to anything outside of itself.

So your perfect circles do have a perfect aspect but they are not absolutely perfect.

CM, any circle has a specific radius, thus the circle is a mathematical ideal, there is no one circle. The function specifies any circle which exists as the instantiation thereof. KF

Right, Cantor said, you create a list of all the natural numbers FIRST, then I can tell you which of my numbers you haven’t found a match for. So I can turn it around and say the same thing, YOU create a list of all of your real numbers, and then I will simply show you one by one how I can match up a number of yours with a number of mine.

Since you can’t even start with two from your list however, I guess that makes it impossible-that’s the card trick. I can list two, and you can’t. Its pretty hard to draw diagonals on your list when you don’t even have one.

If I’m right then it is possible for a proper subset to have the same number of elements as the parent set.

LoL! Obviously you don’t know set theory as a proper subset is defined such that what you say is impossible.

AGAIN- IT’S YOUR SCHEME THAT IS BEING DEBATED. That means you cannot just keep trotting it out as support for itself.

And you cannot justify using your scheme for the reasons provided.

Don’t debate it, find a mistake.

LoL! Talk about being ignorant of how to debate something! I found the mistake, Jerad. And all you can do is use your mistake-ridden formula to “rebut” what I found.

It doesn’t hold for infinite sets.

So you say yet cannot prove.

And it is very telling that Jerad cannot show ONE practical application that refutes my claims and supports his.

kf, think of the essential nature of the ideal circle. The ratio of the radius to the circumference is unchanging, it is an absolute value. No matter what size a physical circles are they will all be an equal approximation of the ideal, everything else being equal. Magnitude is relative and relativity has nothing to do with the ideal. When defining a circle, there are no text books where size is considered. Size only comes into play when we are considering the physical world.

Mathematics is a useful concept and it isn’t philosophy. Bad things tend to happen when we get the math wrong.

Will trajectories be affected if we all agreed that the cardinality of countably infinite sets could be different? Would relativity be refuted? Would the climate ease to change?

What calculations would be affected?

That is if we allowed the standard and accepted definitions and rules of set theory to hold universally, as universal laws do (see the math), meaning ALL proper subsets will have fewer elements than their “parent” or super set, what, in the real world, would be affected? What would be the effect of such a thing?

We could then use the “mapping formula” to tell us the relative difference. For example:

Let set A = {1,2,3,4,…}
Let set B = {2,4,6,8,…}

Would be a relative difference of 2n, meaning set A is twice the size of set B.

And one more time- If Jerad’s methodology was correct I would not be able to use standard set mathematics to prove him wrong. And yet I have.

How did I prove Jerad wrong? By using basic set subtraction I was able to demonstrate That I can remove each and every naturally matched element and still have an infinite set left. And that Jerad refuses to deal with that proves he has nothing but to keep repeating his refuted tripe.

I am not denying your matching gives a different result but you can’t say yours is right and mine is wrong without finding something wrong with mine.

I did. Yours is NOT natural. Yours is a false relationship. Also you cannot say that yours is right and mine is wrong without finding something wrong with mine as I did to yours.

MT, a wheel is close enough to ideal for govt work, but it is never quite perfect

CharlieM @ 550

No true circles exist in the material world.

A perfect circle can’t exist in any world – not possible even in Mathematical world, since Pi can’t be computed fully ( as it is irrational). It is ridiculous to expect a perfect wheel. If there is a perfect circle, then Pi will become rational.
A perfect wheel will, in fact, be more bumpy than a ‘real world imperfect’ circular wheel, as we can never have a perfect plane. At microscopic level, even a ‘perfect’ plane is just a series of cantenary shape. The centroid of a perfect wheel will never follow a straight line if the wheel is a perfect circle hence the ride will be bumpy even if the road is a ‘perfect’ plane.

Right, Cantor said, you create a list of all the natural numbers FIRST, then I can tell you which of my numbers you haven’t found a match for. So I can turn it around and say the same thing, YOU create a list of all of your real numbers, and then I will simply show you one by one how I can match up a number of yours with a number of mine.

But it’s impossible to create a list of all the real numbers so why should I try?

Besides you’re the one making a claim that runs counter to accepted results. You make a list.

Since you can’t even start with two from your list however, I guess that makes it impossible-that’s the card trick. I can list two, and you can’t. Its pretty hard to draw diagonals on your list when you don’t even have one.

It’s impossible to make a list of the reals, that’s what Cantor said. I know that no matter what list I made you could easily match an integer to everything on the list but my list would always be incomplete. So, your game is a non-starter.

Any list of real numbers will be incomplete, that’s the point. That’s what Cantor showed.

“A set is infinite if and only if it is equivalent to one of its proper subsets.”

Welcome to the infinite world.

AGAIN- IT’S YOUR SCHEME THAT IS BEING DEBATED. That means you cannot just keep trotting it out as support for itself.

And you cannot justify using your scheme for the reasons provided.

You haven’t got any reasons. You said it was ‘contrived’ but you couldn’t back that up with a referenced definition. You don’t like my scheme ’cause it proves my point.

Find a documented reason why my scheme is wrong or questionable. You haven’t so far.

Besides, ti’s dead simple.

LoL! Talk about being ignorant of how to debate something! I found the mistake, Jerad. And all you can do is use your mistake-ridden formula to “rebut” what I found.

You found no mistakes. You said it was contrived but couldn’t back up that claim. YOU think the only matching that’s possible is one that matches up identical elements but that’s not true so it’s not a mistake.

So you say yet cannot prove.

I already have.

And it is very telling that Jerad cannot show ONE practical application that refutes my claims and supports his.

Funnily enough that doesn’t matter when you’re talking pure mathematics. Besides, I don’t bump into too many infinite sets in the real world.

So in your worldview existence is conditioned on complete computational reduction.
That explains a lot. I hope you know that there are lots of worldviews that are not so constrained.

Reality is different from an ideal world view. There can be no real worldview which is not constrained in some parameters.

Jerad, If I can remove all of the elements of the proper subset and still have elements left then it is obvious the two sets are not the same size. I am using normal set math. You are using imaginary relationships. That is the mistake I found, Jerad. Your willful ignorance is not a refutation.

And obviously you are also ignorant of mathematical proof.

YOU think the only matching that’s possible is one that matches up identical elements but that’s not true so it’s not a mistake.

LoL! Wrong again, Jerad. The only matching that counts is the natural matching. all others are manufactured and show a false relationship.

OK so there isn’t any practical application and that means anyone can say anything they want about this part of set theory and no one will be able to prove them wrong.

Mathematics is a useful concept and it isn’t philosophy. Bad things tend to happen when we get the math wrong.

Will trajectories be affected if we all agreed that the cardinality of countably infinite sets could be different? Would relativity be refuted? Would the climate ease to change?

What calculations would be affected?

That is if we allowed the standard and accepted definitions and rules of set theory to hold universally, as universal laws do (see the math), meaning ALL proper subsets will have fewer elements than their “parent” or super set, what, in the real world, would be affected? What would be the effect of such a thing?

We could then use the “mapping formula” to tell us the relative difference. For example:

Let set A = {1,2,3,4,…}
Let set B = {2,4,6,8,…}

Would be a relative difference of 2n, meaning set A is twice the size of set B.

And one more time- If Jerad’s methodology was correct I would not be able to use standard set mathematics to prove him wrong. And yet I have.

How did I prove Jerad wrong? By using basic set subtraction I was able to demonstrate That I can remove each and every naturally matched element and still have an infinite set left. And that Jerad refuses to deal with that proves he has nothing but to keep repeating his refuted tripe.

We could then use the “mapping formula” to tell us the relative difference. For example:

Let set A = {1,2,3,4,…}
Let set B = {2,4,6,8,…}

Would be a relative difference of 2n, meaning set A is twice the size of set B.

Except those two sets are the same size.

And one more time- If Jerad’s methodology was correct I would not be able to use standard set mathematics to prove him wrong. And yet I have.

Nope, see the theorem (that means it’s been proved) I referenced above which says that a set is infinite if and only if it is equivalent to one of its proper subsets.

Welcome to the infinite world.

How did I prove Jerad wrong? By using basic set subtraction I was able to demonstrate That I can remove each and every naturally matched element and still have an infinite set left. And that Jerad refuses to deal with that proves he has nothing but to keep repeating his refuted tripe.

Because I found a 1-to-1 matching while your subtraction is not a 1-to-1 matching. It’s pretty simple really.

I did. Yours is NOT natural. Yours is a false relationship. Also you cannot say that yours is right and mine is wrong without finding something wrong with mine as I did to yours.

Not natural? What does that mean? You couldn’t back up contrived so now your saying my matching is not natural? It’s a false relationship? What are you talking about?

I didn’t say yours was wrong, it’s just not 1-to-1. There are lots and lots and lots of ways to match up the elements of those two sets. I found one that’s 1-to-1 which can only be true if the sets are the same size.

I really don’t understand this notion that there’s only one (natural?) way to match up elements of two sets? Can you please show me a set theory book or article which says that?

Zachriel: A circle is the set of points equidistant from a center point.

mike1962: Right. An infinite set of points. Thus nonsense…

Heh. It’s the definition.

kairosfocus: Pi is worse, it is transcendental

We prefer to think of it as better.

phoodoo: YOU create a list of all of your real numbers, and then I will simply show you one by one how I can match up a number of yours with a number of mine.

As Cantor proved, no one can list the real numbers.

Jerad, If I can remove all of the elements of the proper subset and still have elements left then it is obvious the two sets are not the same size. I am using normal set math. You are using imaginary relationships. That is the mistake I found, Jerad. Your willful ignorance is not a refutation.

Nope. Take the natural numbers 1, 2, 3, 4 . . .

Tale out the infinite set 2, 4, 6, 8 . . .

You’ve got left 1, 3, 5, 7 . . .

All three of those sets are the same size. And I can prove it by finding a 1-to-1 correspondence between any pair of them.

You need to stop using finite notions and start using infinite ones.

I didn’t make a mistake, you’re not working with infinite sets properly. I am not dodging an issue,

And obviously you are also ignorant of mathematical proof.

Really? Well, the theorem I linked to had three proofs associated with it. If you can find a mistake in any of them I’ll reconsider.

LoL! Wrong again, Jerad. The only matching that counts is the natural matching. all others are manufactured and show a false relationship.

A contention which you have not yet referenced or justified. Please do.

OK so there isn’t any practical application and that means anyone can say anything they want about this part of set theory and no one will be able to prove them wrong.

Not at all. A lot of people thought the Axiom of Choice was obviously true but it ain’t. That’s why it’s called an Axiom now, even though it’s equivalent to Zorn’s lemma which is confusing really. And it’s equivalent to the Well Ordering Principle. The names are historical hold-overs.

Look Joe, if you’re right then you should be able to find set theoretical definitions of ‘contrived’ and ‘natural’ regarding mappings (which is the way mathematicians referring to the kind of matchings I’m doing).

Think about a function . . . say f(x) = 3x – 1

Put in an x, say x = 3, get out a result, in this case 8. The function matches 3 with 8. AND, if you draw a picture (graph) all the pairs that the function ‘creates’ you get a straight line. A function is a matching between two sets, normally called the domain and the range. This function is 1-to-1 because every element in the domain (the x-values) is matched which exactly one result or outcome (the y-values when we graph the function on the (x,y) plane).

The function I’ve just listed is the same as the matching scheme I used between my two sets. My matching is a 1-to-1 function between the two sets. It tells you how to get from an element in one set to its unique counterpart in the other set.

I am not trying to be belligerent nor am I making absurd claims. The mathematics I’m referencing and the techniques I’m using are not controversial or complicated. The theorem I linked to (A set is infinite if and only if it is equivalent to one of its proper subsets) is not new or made up. And there’s three proofs on the web page.

Anyone can look up set theory on Wikipedia, read and follow some links and find most of this out. It’s not just me saying these things.

I don’t understand why some people are having such a hard time with it.

fifthmonarchyman: Can a worldview be real if it can not be computed fully?

There are infinitely many noncomputable numbers*. While we might speculate the universe is discrete and computable, that is not necessarily the case. Reality doesn’t have to stoop to our limited expectations.

* Computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers or the computable reals.

Jerad, If you were right then we wouldn’t be able to do what I said and still have an infinite set left. That proves that yours is a manufactured scheme meant to fool people.

Obviously it has worked on you and others. Don’t blame us because you are easily fooled.

What exactly is a “real” worldview?
Can a worldview be real if it can not be computed fully?

The world that one faces when s/he is awake and sober is the Real Worldview.
Any world view can be computable only up to a level that we understand it, so computability is progressive – we can compute more today than a few decades back, and we can compute more a decade from now.

Any world view can be computable only up to a level that we understand it, so computability is progressive – we can compute more today than a few decades back, and we can compute more a decade from now.

I say,

Just like Pi

Some worldviews allow us to continue to progress in understanding infinitely.

Others are constrained to only understanding the finite phyiscal universe.

Jerad, If you were right then we wouldn’t be able to do what I said and still have an infinite set left. That proves that yours is a manufactured scheme meant to fool people.

Huh? The natural numbers are countably infinite. The even natural numbers are also countably infinite. The odd natural numbers are countably infinite. You can take an infinite set away from another infinite set and still have an infinite number of things left. Obviously.

Obviously it has worked on you and others. Don’t blame us because you are easily fooled.

Have you looked at that theorem or found a reference for your use of the term contrived or have you found a reference for your claim that only the ‘natural’ mapping/matching is right?

I can’t blame anyone for finding dealing with infinities weird the first few times. Which is why I think it’s good to do some reading and see what’s already been done first. It’s easy these days!!

phoodoo: If the real numbers can’t be listed, how can you make a one to one correspondence?

It’s provable that you can’t make a one-to-one correspondence between the natural numbers and the real numbers. However, it’s provable that you can make a one-to-one correspondence between the natural numbers and the rational numbers.

A perfect circle can’t exist in any world – not possible even in Mathematical world, since Pi can’t be computed fully ( as it is irrational). It is ridiculous to expect a perfect wheel. If there is a perfect circle, then Pi will become rational.
A perfect wheel will, in fact, be more bumpy than a imperfect circular wheel, as we can never have a perfect plane. At microscopic level, even a ‘perfect’ plane is just a series of cantenary shape. The centroid of a perfect wheel will never follow a straight line if the wheel is a perfect circle hence the ride will be bumpy even if the road is a ‘perfect’ plane.

MT, I don’t much care for labels, but: I presume you are a materialist. I would call myself an objective idealist.

If you think about it a perfect circle can’t exist in the physical world. From your comments you are giving us an example of a perfect circle in the physical world. Well no one is arguing that this is a possibility.

Why do you think that Pi needs be rational? It might conform to neatness and tidiness according to our human minds, but what has that got to do with anything? The fact that Pi stretches on to infinity is no problem for the ideal world. It is the world of infinities and absolutes.

You say the perfect circle can’t exist in any world implying that you know all worlds. The ideal circle which I say is real will be the same circle throughout all of time, its definition will not change, but any wheel you care to mention had a beginning and it will have an end. In other words my circle is permanent but yours is transient. But you still say that yours has more reality than mine. Obviously I disagree.

None of Jared’s posts are his own personal opinion or unconventional. Every point he has made is standard mathematical infinite set theory that can be found on the internet or in any textbook on the subject.

.

As for Joe, so far I’ve been unable to discern whether

a) He is not familiar with (or doesn’t understand) infinite set theory,

or

b) He understands infinite set theory but simply rejects it.

.

If it’s (a), that’s easily resolved by doing a bit of homework.

If it’s (b), then Joe is certainly free to make up his own mathematics. But it would be helpful if he would clearly state that’s what he’s doing, and rigorously define the terms he uses.

We could then use the “mapping formula” to tell us the relative difference. For example:

Let set A = {1,2,3,4,…}
Let set B = {2,4,6,8,…}

Would be a relative difference of 2n, meaning set A is twice the size of set B.

Ok, so based on this, set A would be thrice the size of {3,6,9,12,…}, five times the size of {5,10,15,20,…} and 2.33333… times the size of {7/3, 14/3, 7, 28/3, …}.

Got it.

But wait – what about subsets that can’t be formed this way? No problem, says Joe – just take the function from which they’re derived, and divide by n. Easy.

So, for example, {1,2,3,4,…} is n times the size of {1,4,9,16,…}. Joe himself confirms this:

The relative difference in cardinality between the two sets would be n squared.

Except… n isn’t a number. Saying one of these sets is n times the size of the other makes no sense at all. Does their difference in size increase? Is it as large as you want it to be?? Could it be infinite???

Further examination of Joe’s ‘theory’ produces more peculiar consequences. Presumably {1,2,3,4,…} is 2n times as large as {2,8,18,32,…}; n^2 times as large as {1,8,27,64,…}; and 1/n larger than {2,3,4,5,…}.

But what about these sets?
{1,4,27,256,…}
{1,1/2,1/3,1/4,…}
{1,sqrt(2),cbrt(3),sqrt(sqrt(4)),…}

Standard set theory has no problem at all, but Joe’s ‘theory’ explodes in a frenzy of leaking grey matter.
However, all this can be cleared up simply by adopting one further theory: Joe is an incompetent buffoon.

Why do you think that Pi needs be rational?…..You say the perfect circle can’t exist in any world implying that you know all worlds.

I don’t have to know all worlds. If you need a perfect circle, Pi has to be rational, which is not possible so you can’t have a ‘perfect circle in any world.

In other words my circle is permanent but yours is transient. But you still say that yours has more reality than mine.

An approximated circle is what works in the world that you face everyday, that is what is used to engineer your car’s wheels – that is reality.
IMO, there is no ‘ideal world’ where ‘circles are perfect and permanent’. Face the reality.

You need to stop using finite notions and start using infinite ones.

But then this thread would not go on ad infinitum…

“Within Cantor’s lifetime, logicians demonstrated that set theory is frankly inconsistent. Dangers in mathematics do not get more dangerous than this.”

Me_Think: If you need a perfect circle, Pi has to be rational, which is not possible so you can’t have a ‘perfect circle in any world.

Not sure why that follows. A circle can be drawn with a string or compass without reference to arithmetic of any sort. It has more to do with lack of perfection in the instrument. http://momath.org/wp-content/uploads/compass.jpeg

MT, You have stated that Pi has to be rational but you haven’t said why it has to be so. Why does it have to conform to your idea of perfection which in fact only relates to simplicity? Simplicity and perfection are two different concepts. It would be much simpler if Pi=3 but that would not conform to reality.

In your opinion there is no ideal world, in my opinion there is. You are right that the world of our normal experience is transient but that is because of the way we are constituted and not necessarily because that is the way it truly is.

This thread is populated mostly with comments from cowards, liars, fruitcakes, malignant narcissists and butt kissers. It makes my blood boil. See y’all around. 😀

PS. Sorry KF. I’m pointing fingers at no one in particular. The above is just a general observation. “If the shoe fits” and all that jazz.

An axiom or postulate is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.

…

In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems.

Not sure why that follows. A circle can be drawn with a string or compass without reference to arithmetic of any sort. It has more to do with lack of perfection in the instrument.

I agree, but they are talking in terms of irrationality of Pi. I am just arguing that since Pi is irrational, there can’t be a ‘perfect circle’ in the sense that circumference will differ – Eg: For radius 2, at 4 digit precision, circumference will be 12.5664, at 20 digit 12.566370614359172954, at 24 12.5663706143591729538506, at 28 digits precision 12.56637061435917295385057353 and so on.

MT, You have stated that Pi has to be rational but you haven’t said why it has to be so. Why does it have to conform to your idea of perfection which in fact only relates to simplicity?

Please see my comment to zac (@595)

You are right that the world of our normal experience is transient but that is because of the way we are constituted and not necessarily because that is the way it truly is.

Me_Think: I am just arguing that since Pi is irrational, there can’t be a ‘perfect circle’ in the sense that circumference will differ – Eg: For radius 2, at 4 digit precision, circumference will be 12.5664, at 20 digit 12.566370614359172954, at 24 12.5663706143591729538506, at 28 digits precision 12.56637061435917295385057353 and so on.

No. We don’t have to measure or calculate pi to make a perfect circle. We don’t have to know if pi is close to three or three and a bit, or even if we don’t have arithmetic at all. If we had a perfect compass and plane, then our circle would be perfect, even if our measurement of the circumference were not.

Me:You are right that the world of our normal experience is transient but that is because of the way we are constituted and not necessarily because that is the way it truly is.

MT:How do you know it is not true ?

Well I can’t say definitely that it isn’t true but to say that what humans experience is a complete view of reality is to say that we are omniscient. You either believe this or you are being inconsistent.

None of Jared’s posts are his own personal opinion or unconventional. Every point he has made is standard mathematical infinite set theory that can be found on the internet or in any textbook on the subject.

Thank you for that.

phoodoo #583

There are so many responses to show why that is baloney, but here is an easy one: Wikipedia says Richard Dawkins knows what he is talking about .

Wikipedia is a propaganda platform for “scientific skeptics” who are never skeptical.

Get any introductory set theory book then. I was suggesting Wikipedia because it’s easy and free.

Mung #588

“Within Cantor’s lifetime, logicians demonstrated that set theory is frankly inconsistent. Dangers in mathematics do not get more dangerous than this.”

– David Berlinksi

I’ll stick with the majority of mathematicians thanks. It would be nice to have a bit more context for the Dr Berlinski comments reproduced. Since I don’t own any of his books it would be nice if someone could give a bit more of the surrounding text.

Mapou #593

This thread is populated mostly with comments from cowards, liars, fruitcakes, malignant narcissists and butt kissers. It makes my blood boil. See y’all around.

It’s so nice to be appreciated and respected.

I’m used to being told my opinions of evolution are wrong but I am somewhat taken aback that established, non-controversial mathematics can cause such a fuss.

It’s sad to that with the exception of KF, none of the UD regulars have bothered to either support the precepts of set theory or correct their fellows. Oh well, it’s not my blog, I don’t make the rules or police the participants.

From Dr Dembski’s 2005 paper: Specification: the pattern that signifies intelligence, page 5, second paragraph (I’ve added some bold emphasis):

“To see this, consider that a reference class of possibilities ?, for which patterns and events can be defined, usually comes with some additional geometric structure together with a privileged (probability) measure that preserves that geometric structure. In case ? is finite or countably infinite, this measure is typically just the counting measure (i.e., it counts the number of elements in a given subset of ?; note that normalizing this measure on finite sets yields a uniform probability). In case ? is uncountable but suitably bounded (i.e., compact), this privileged measure is a uniform probability. In case ? is uncountable and unbounded, this privileged measure becomes a uniform probability when restricted to and normalized with respect to the suitably bounded subsets of ?. Let us refer to this privileged measure as U.”

“Although the combinatorics involved with the multinomial distribution are complicated (hence the common practice of approximating it with continuous probability distributions like the chi- square distribution), the reference class of possibilities ?, though large, is finite, and the cardinality of ? (i.e., the number of elements in ?), denoted by |?|, is well-defined (its order of magnitude is around 1033).”

“This sequence, as well as the totality of such agents, is at most countably infinite. Moreover, within the known physical universe, which is of finite duration, resolution, and diameter, both the number of agents and number of patterns are finite.”

“Strictly speaking, P(.|H) can be represented as f.dU only if P(.|H) is absolutely continuous with respect to U, i.e., all the subsets of ? that have zero probability with respect to U must also have zero probability with respect to P(.|H) (this is just the Radon-Nikodym Theorem”

So I guess Dr Dembski ‘believes’ in countably infinite sets and continuous function.

It’s sad to that with the exception of KF, none of the UD regulars have bothered to either support the precepts of set theory or correct their fellows.

I say.

Off the top of my head I remember Myself, Mung, ba77, and CharlieM all defending the reality of infinity here and your fellow traveler Me_think actively challenging the concept.

Mung: Lets’ agree that we’re talking about faith here, can we?

Zachriel: No. Were talking about what can be deduced from given axioms. What axioms in ZFC do you reject?

Which definitions of axiom do you reject?

I think that “faith” is simply the wrong word here. You don’t choose axioms based on whether you believe they are absolute truths or not. It’s more of a question of whether the resulting mathematics is interesting and/or useful.

For example, the somewhat nonintuitive axiom “all triangles have angle sum strictly less that 180 degrees” turns out to be quite productive in certain contexts.

Off the top of my head I remember Myself, Mung, ba77, and CharlieM all defending the reality of infinity here and your fellow traveler Me_think actively challenging the concept.

That you did not notice any of this is telling.

I was talking about the precepts of set theory not just the notion or reality of infinity. And I’m quite sure you misread Me_thinks comments.

But let’s just get to the meat of the matter: do you agree with the notion that you can have a countable infinite set (like the natural numbers), take out a countably infinite set (like the even numbers) and still have a countably infinite set left (the odds in this case)?

Why not just commit on the mathematics and get past the he-said, you-said stuff. Where do you fall on the mathematical fence?

I have told you: mathematics of infinite sets works differently.

I know and I am pretty sure that is part of the claim we are debating. So just repeating it proves that you are senile.

{1,2,3} a finite set that has the first three positive integers. {1,2,3,…} is an infinite set of positive integers, the pattern which was directly extrapolated from the established finite pattern. There isn’t any difference in mathematics there.

OK how about set subtraction? That seems to be work the same. Is subtraction still part of mathematics?

No, Jerad, the only difference seems to be is that with infinite sets Cantor manufactured a mathy-sounding solution and most people seem to have bought it. Unfortunately it doesn’t do them a world of good but they think it makes them smarter than others cuz they know the troof ’bout infinity stuff by golly.

Please wake us when you find a practical application so we can actually test this concept to see who is right.

In fact I’ve argued in other places that Cantor’s theorem is the reason that you won’t be able to write an algorithm that will fool the observer in my game.

Now I do find the question of whether infinity actually exists to be more interesting than discussions of cardinality. So I don’t spend a lot of time defending Cantor around materialists

Do you think that infinity actually exists or is it just an imagination of the human mind that for some reason just happens to be useful in sophisticated math?

If infinity is just human imagination then discussions of cardinality are “chasing after the wind”.

I know and I am pretty sure that is part of the claim we are debating. So just repeating it proves that you are senile.

But you keep repeating your finite version of set mathematics.

Take the natural numbers. Take out all the multiples of 2 starting with 4. That’s taking out an infinite set. But you’ve still got the odds plus 2 and that’s still infinite. Take out all the multiples of 3 starting with . . . 6 is already gone . . . starting with 9. That’s taking out an infinite number of elements but you’ve still got an infinite number of things left. But when I take out elements from a set it can’t stay the same size according to you so it must be getting smaller.

You want there to be an infinite amount of infinities getting smaller and smaller and smaller. With no lower bound. And never crossing over into the finite. And that’s been shown to be unworkable. Go read an introductory book on set theory. I shouldn’t have to type a whole textbook into this blog just for you.

{1,2,3} a finite set that has the first three positive integers. {1,2,3,…} is an infinite set of positive integers, the pattern which was directly extrapolated from the established finite pattern. There isn’t any difference in mathematics there.

Mathematics with infinite sets works differently. And you can easily learn about this by LOOKING IT UP.

OK how about set subtraction? That seems to be work the same. Is subtraction still part of mathematics?

Yes but it works differently with infinite sets.

There were so many concepts and ideas and things which challenged me and really made me think when I took mathematics courses. It wasn’t spiritual, it wasn’t philosophical it was . . . trying everything out and seeing where it went.

No, Jerad, the only difference seems to be is that with infinite sets Cantor manufactured a mathy-sounding solution and most people seem to have bought it. Unfortunately it doesn’t do them a world of good but they think it makes them smarter than others cuz they know the troof ’bout infinity stuff by golly.

Uh huh. Joe, why can’t you support your view of set theory mathematics with some references? Why haven’t you addressed the theorem I linked to? Why do you use terms in non-standard ways without giving some examples, at least, of why you’re justified in doing so.

Please wake us when you find a practical application so we can actually test this concept to see who is right.

If your life is only going to be defined by what is practical then it’s going to be very flat and dull in my mind. Can you point to a practical application that comes out of believing in a cosmic designer? Can you show me some practical reason for reading Shakespeare or Dante or Melville or Poe or Hawthorne? Can you give me a practical reason for viewing the paintings of Raphael, Titian, Caravaggio, Vigee Lebrun, Pollack, Magritte? Why should we watch films like 2001 or The Day the Earth Stood Still or Syriana? Why should we care about black holes or CMEs or exo-planets or comets? None of this stuff affects us on a day=to-day basis. Let’s just throw it all out and spend our time and money on practical things.

To paraphrase KF again: God is a kick-ass mathematician. As you would expect.

I would very much like to hear the reasoning behind your alias.

Cantor’s side. That should be obvious.

I was just checking.

In fact I’ve argued in other places that Cantor’s theorem is the reason that you won’t be able to write an algorithm that will fool the observer in my game.

I’m not sure that follows but . . . okay.

Now I do find the question of whether infinity actually exists to be more interesting than discussions of cardinality. So I don’t spend a lot of time defending Cantor around materialists

What does materialism have to do with it? We’re talking pure mathematics!!

Honestly, I don’t understand why this is coming up at all.

Do you think that infinity actually exists or is it just an imagination of the human mind that for some reason just happens to be useful in sophisticated math?

What is sophisticated math? Do you mean anything over a high school level? ‘Cause there’s a lot of math past that.

I think that we cannot experience infinity as a physical reality. But IF you’re going to do mathematics then you damn well better comes to terms with it, knock it about a bit and wrestle it to the ground. Because otherwise you end up just standing at the edge of a cliff (up or down?) not moving.

If infinity is just human imagination then discussions of cardinality are “chasing after the wind”.

I think any intelligent creature who starts down the mathematics path is going to come up against these same issues. I think the questions are universal.

What’s absolutely incredibly amazing is that, within certain parameters and given certain axioms, mathematicians have come up with ways of thinking about infinities that are consistent and useable. Not useable as we experience driving to work or playing with the kids or paying bills. But useable in the exploration of the mathematical landscape.

Honestly, I don’t understand why this is coming up at all.

I say,

If infinity is not real you have just spent over 600 comments having a “how many angels on the head of a pin” discussion

if infinity is just a synaptic buzz in the human brain there is literally no way to demonstrate Mapou and Joe are incorrect.

If you want to prove them and those like them wrong you need to demonstrate that things outside the cave have a real existence.

I just don’t think you can do that given your presuppositions

You say,

What’s absolutely incredibly amazing is that, within certain parameters and given certain axioms, mathematicians have come up with ways of thinking about infinities that are consistent and useable.

I say,

You do know that my worldview has an explanation for that phenomena don’t you?

To paraphrase KF again: God is a kick-ass mathematician. As you would expect.

I say

Exactly

quotes:

That God is of himself, that is, neither from another, nor of another, nor by another, nor for another But is a Spirit, who as his being is of himself, so he gives being, moving, and preservation to all other things, being in himself eternal, most holy, every way …….infinite…….. in greatness, wisdom, power, justice, goodness, truth, etc.

Jerad quote: “I’ll stick with the majority of mathematicians thanks.”

This is not erudition, He is simply displaying typical ‘scientific skeptic” cult behavior. They pretty much run science departments in this country if you don’t know.

One of their propensities is to get together and say, “Ok, these are the things we need to believe in to maintain our materialist worldview, which is critical. Under NO circumstances are they to be questioned, because then we will have to be forced to ponder the world we live in. That is unacceptable! Neil Degrasse Tyson, Jerry Coyne, Seth Shostak, Steven Novella, Richard Dawkins, Danniel Dennet, Evan Bernstein, Bill Nye, Penn Jillette, Ira Flatow, Lawrence Krauss, Jad Abumrad, get out there and spread the word. Here is the list of the things we believe in…..

Be unified. It is imperative that no one waivers even a fraction, regardless of any so called evidence. We must be strong! Oh, and call yourself a skeptic for crying out loud, the irony helps to confuse people!!”

Keiths- (A member of the erudite “scientific skeptic” cult.)

“I, I , I need to stop this discussion now! It’s in direct confrontation with my worldview of never questioning materialism, or the founder of our great movement, the plagiarist Darwin, and his modern day disciples. But, but, I have nothing to say, I don’t even really know what mathematics is…what can I do?? Oh well, I will just fall back on my usual strategy, just bang my head on the keyboard and hope it strikes some keys. It always works for me in the past. I don’t see why I always get banned everywhere, I can’t help what my forehead hits.”

If infinity is not real you have just spent over 600 comments having a “how many angels on the head of a pin” discussion

if infinity is just a synaptic buzz in the human brain there is literally no way to demonstrate Mapou and Joe are incorrect.

If you want to prove them and those like them wrong you need to demonstrate that things outside the cave have a real existence.

I just don’t think you can do that given your presuppositions

Well Dr Dembski clearly understands and uses countably infinite sets and he teaches divinity now I believe.

Also I cannot force someone to explore and question their own beliefs. I cannot summarise whole textbooks of materials on a blog. I have tried to explain and answer questions to the best of my abilities.

Aurelio Smith #617

I think Jerad deserves a round of applause for his patience and erudition (I particularly appreciate his citations of William Dembski; see 602 – 605)

Thank you. I keep thinking of things I could have said better! Ah well.

Aurelio is such a lovely name.

Cantor #619

ditto that

Thank you. Your alias is very comforting at least!!

phoodoo #620

This is not erudition, He is simply displaying typical ‘scientific skeptic” cult behavior. They pretty much run science departments in this country if you don’t know.

One of their propensities is to get together and say, “Ok, these are the things we need to believe in to maintain our materialist worldview, which is critical. Under NO circumstances are they to be questioned, because then we will have to be forced to ponder the world we live in. That is unacceptable! Neil Degrasse Tyson, Jerry Coyne, Seth Shostak, Steven Novella, Richard Dawkins, Danniel Dennet, Evan Bernstein, Bill Nye, Penn Jillette, Ira Flatow, Lawrence Krauss, Jad Abumrad, get out there and spread the word. Here is the list of the things we believe in…..

Be unified. It is imperative that no one waivers even a fraction, regardless of any so called evidence. We must be strong! Oh, and call yourself a skeptic for crying out loud, the irony helps to confuse people!!”

Since you have already made up your mind I shan’t bother to try and discuss these issues with you anymore. But I would strongly suggest you get an introductory book on set theory and read it with as open a mind as possible. It is a beautiful and mind stretching field of mathematics. Personally I prefer number theory but when in Rome . . .

phoodoo #622

“I, I , I need to stop this discussion now! It’s in direct confrontation with my worldview of never questioning materialism, or the founder of our great movement, the plagiarist Darwin, and his modern day disciples. But, but, I have nothing to say, I don’t even really know what mathematics is…what can I do?? Oh well, I will just fall back on my usual strategy, just bang my head on the keyboard and hope it strikes some keys. It always works for me in the past. I don’t see why I always get banned everywhere, I can’t help what my forehead hits.”

You have a very strange view of some of your fellow human beings.

I am just arguing that since Pi is irrational, there can’t be a ‘perfect circle’ in the sense that circumference will differ – Eg: For radius 2, at 4 digit precision, circumference will be 12.5664, at 20 digit 12.566370614359172954, at 24 12.5663706143591729538506, at 28 digits precision 12.56637061435917295385057353 and so on.

In the above statement you are saying that a circle will expand or contract depending on how accurate Pi is expressed. The circumference will not differ. The ratio of the radius to the circumference of a circle is fixed for eternity, it will never change. What has changed above is your precision in expressing Pi. Pi is not the variable, you are.

The circumference will not differ. The ratio of the radius to the circumference of a circle is fixed for eternity, it will never change.

The values given are the circumference of the circle. Circumference will differ as Circumference is 2*Pi*r. Since circumference differs, ratio of radius to circumference too will differ (hence it can’t be a ‘perfect circle’).
I think it would be better if you define what you mean by a ‘Perfect Circle’.
P.S: If you mean Pi’s precision is fixed before hand, then of course, circumference and ratio will not change, but what is the ‘perfect precision’ for the circle ? Is there a limit to the precision which will satisfy the definition of perfect circle – assuming you have one?

MT, You have it backwards. Pi does not determine the form of the circle, it is derived from the form. Pi is derived from the circumference divided by the diameter. This value will never change regardless of our ability to measure it precisely.

Why can it not be measured precisely?

Well if we could somehow obtain a piece of thread with zero thickness that was exactly the same length as the diameter and we laid it exactly on the circumference, we could measure three lengths but there would still be part of the circumference left over. So we divide our thread into ten equal pieces. We could then try to bridge the gap left but there would still some left over. So we further divide the thread we are using into ten equal pieces. We find that four pieces will almost bridge the gap but we still haven’t reached our starting point. No matter how many times we divided our thread in this way we would never reduce the gap to zero.

In a perfect circle the distance from the centre to the circumference with always remain the same no matter what point of the circumference the radius touched.

LoL! @ Jerad- Dembski did NOT use the concept tat all countable and infinite sets have the same cardinality. THAT is what we have been debating.

Also, Jerad, by your “logic” Cantor was wrong because he couldn’t reference anyone already using his concept!

Jerad sed that the mathematics of infinite sets is different yet I have provided examples that refute his claim. So what does he do? Ignore it and prattle on.

I know and I am pretty sure that is part of the claim we are debating. So just repeating it proves that you are senile.

Jerad:

But you keep repeating your finite version of set mathematics.

I don’t have a finite version of set mathematics. However I will repeat my refutation of your concept every time you repeat that concept. It only takes ONE refutation, meaning I can repeat the one that works until you face it.

I know and I am pretty sure that is part of the claim we are debating. So just repeating it proves that you are senile.

Jerad:

But you keep repeating your finite version of set mathematics.

I don’t have a finite version of set mathematics. However I will repeat my refutation of your concept every time you repeat that concept. It only takes ONE refutation, meaning I can repeat the one that works until you face it.

Consider a hotel with an infinite number of rooms a room for each of the natural numbers, and all the rooms are filled. A new guest arrives. We can move the guest in room 1 to room 2, the guest in room 2 to room 3, the guest in room N to room N+1, and so on. Now room 1 is open for the new guest. We can do this again and again. The Grand Hotel is always full, yet always has room for more.

Folks, passing by briefly. As already shown, the naturals, evens, odd (and more) are countable and for each there is no last such number, there is always one more, then another endlessly. That yields a common cardinality, aleph-null, which takes the just outlined meaning. To try to extend to such a situation how we scale finite countable sets will fail. Beyond, we can define further sets such as the set of subsets of naturals, which will have scale 2^aleph-null. This may plausibly be seen as the continuum number as it takes in all ordered or structured n-tuples [just suitably assign meanings to the list of any given subset per decimal place value assignments and the requisites of co-ordinate systems X,Y, r-theta, x1, jx2, etc] such as would specify co-ordinates of points in the plane or 3-d space or any ball in a vector space of desired dimension] etc. Of course that is not proved. KF

Zachriel:Consider a hotel with an infinite number of rooms a room for each of the natural numbers…

Joe: A new natural number is born! I told you this was magical stuff!

Hehe.

It would seem to me that…

Nonsense + anything else = nonsense

Physicists hate infinity. They do their best to “cancel infinities out.” Which tells me what it’s really all about is symbolic manipulation where “infinities” are merely placeholders for nonsense which seems to conform to something, but nobody know what “it” is, and stick a label of “infinity” to it.

KF: Folks, passing by briefly. As already shown, the naturals, evens, odd (and more) are countable and for each there is no last such number, there is always one more, then another endlessly.

“True” in “theoretical concept”, false in actualization. The problem is, nothing can actually count them. There will always be a disconnect between what is “countable” and a count actually getting accomplished in the real world.

The nonsense comes into play (and why physicists hate infinity) is when you try to actually get real-world meaning out of an “infinite but countable” set. Matters are “worse” with the aleph one set, which is “non-countable”, i.e, no mapping from aleph null to aleph one.

Zachriel: You didn’t answer the question. How many natural numbers are there? Is it finite? Or not finite?

The question wasn’t directed to me, but I will say intuitively it seems the answer is “infinite”, but I don’t really know what that means beyond what seems like an algorithmic definition that has no actualization in reality, and which physical scientists make serious attempts to avoid and “cancel out.”

The nonsense comes into play (and why physicists hate infinity) is when you try to actually get real-world meaning out of an “infinite but countable” set.

Physicists also have to pay close attention to the distinction between the countably and uncountably infinite. Hilbert spaces with countable bases are ubiquitous in QM; Hilbert spaces with uncountable bases are generally avoided for reasons which are mostly beyond my grasp, but my understanding is that the uncountable case introduces some very difficult problems that are best avoided.

mike1962: The question wasn’t directed to me, but I will say intuitively it seems the answer is “infinite”,

Okay.

mike1962: but I don’t really know what that means beyond what seems like an algorithmic definition that has no actualization in reality,

In mathematics, it’s called induction. For each natural number n, there is a natural number n+1. This is also called the axiom of infinity.

mike1962: and which physical scientists make serious attempts to avoid and “cancel out.”

Physicists have often used infinities, such as for solving integrations. See Newton, Mathematical Principles of Natural Philosophy, Philosophical Transactions of the Royal Society 1687.

Zachriel: In mathematics, it’s called induction. For each natural number n, there is a natural number n+1. This is also called the axiom of infinity.

You’re basically restating ground already covered. But thanks.

Physicists have often used infinities, such as for solving integrations. See Newton, Mathematical Principles of Natural Philosophy, Philosophical Transactions of the Royal Society 1687.

Sure, they have gotten “used”, and still do, but they are not allowed to remain in any practical results, since that is nonsense. I’m not saying “infinities” don’t exist in some sense, and are symbolically manipulated, but what I think they really are beyond symbolic and algorithmic definition and the intuition involved in our minds with respect to them, well, I think I already covered that.

M62: Prob is of course, once calculus came in the door so did all of this implicitly. And sans calculus and its extensions, bye bye modern world. I suggest the best stance is to recognise that mathematics addresses an ideal space of forms, with a logical structure, indeed Mathematics can be defined as the logical and sometimes quantifiable study of structure. Then, we can extend and apply mathematical models to the real world, recognising that Plato et al had a point (though not the whole story) when they spoke of a world that imperfectly mirrors the ideal abstract world. Or in terms that are more relevant to us, mathematics is about a common, shared mental-logical space which has powerful relevance to the experienced world of our common physical existence; e.g. that a square circle is logically impossible and therefore physically unrealisable even to approximation (e.g. by bending a paper-clip), is a seemingly simple case with all sorts of subtle but powerful implications that come out on pondering what it points to. Which has in it all sorts of suggestions as to the roots of our world and how it comes to be so ordered. From my view the first thing, almost, that God is, is a Mathematician. KF

PS: I think also that the point on what cardinality of a set is, is that it speaks to scaling of a set, leading to the point that two sets share common cardinality if they may be exactly matched in their members in some orderly way. For sets that are exhaustible in finite steps, that leads to you can count the sets. But when we have sets that are transfinite, we can only set up a set-builder notation or assignment that shows the logical status of that correspondence or otherwise. Once that is done, we can decide whether or no a given thing is in the set and we may assign a match to a standard set such as the natural numbers. It is then a logical demonstration that due to the inexhaustibility, the naturals match the evens and odds, etc, also the rationals. But, once we go over to continuum, we have ways to show that we can exhaust the naturals but will always have further members left over, i.e. the real numbers are in a continuum and have a higher order cardinality. I have already suggested that by looking at the set of subsets of the naturals, we may get a glimpse of how an ordered structure and functional interpretation on such subsets (which BTW will require repetition of digits) will allow us to in principle address every point in a space assignable a coordinate system. So, the continuum can reasonably be seen as arguably having cardinality c = aleph_1. But that is light years away from an actual proof.

I’ve been reluctant to continue commenting on this thread given the dishonesty and gutlessness of some of the commenters but this comment by Zachriel is interesting because it reveals the crux of the problem with infinity.

Zachriel:

For each natural number n, there is a natural number n+1. This is also called the axiom of infinity.

Of course, this does not prove that anybody can build an infinite set or that an infinite set exists anywhere. It’s just a rule that says you can add 1 to any natural number n to obtain a new natural number that is neither n nor 1. The rule is self contradictory for the following reason:

To any natural number n, one can also add ANY natural number such as 2 or 3 or 99 or 2422435, etc. The only exception is infinity. Why? It’s because adding infinity to n does not yield another natural number. It yields, well, infinity.

In other words, if infinity is a natural number that exists, one cannot add a natural number to it to yield a new natural number. The so-called axiom of infinity is broken before it even gets out of the gate. 😀

Joe: There are more natural numbers than there are natural even numbers. It is all relative

Joe, there are just as many even numbers as natural numbers.

Mung, the reason for the disconnect is that Joe is using his own private definition of “more”. He is developing his own infinite set theory. (e.g. See Jared’s post 658 below)

Z, pardon, the cardinality of set N is aleph-null, not something so broad as “infinity.” The issue here is to address inexhaustible sets and do so reasonably in a way that does not toss Calculus. That will lead to what Cantor et al achieved or something very much like it. KF

My reading of your interpretation of set theory is that the cardinality of the natural numbers is bigger than the cardinality of the evens which in turn is bigger than the cardinality of the multiples of 4 which is larger than the cardinality of the multiples of 8, ad infinitum?

Is that correct? All the sets listed above are infinite but their cardinalities are decreasing?

If I continued the process (multiples of 16, then 32, then 64 . . . ) I’d keep getting smaller and smaller cardinalities but I’d still have infinite sets.

Correct?

Is there then a lower bound to the cardinalities or do they continue to decrease without ‘hitting a wall’ so to speak. If there is a lower bound do the cardinalities hit it or do they approach it asymptotically?

I can’t answer this question because this is your version of infinite set theory. Only you can answer it.

Zachriel (completely and dishonestly ignoring my point that, if a size n is not a natural number, it does not exist) retorts:

How many natural numbers are there? Is it finite? Or is it not finite?

The question is self-contradictory because it assumes that which it is trying to prove. It assumes that natural numbers can be fully counted to determine a quantity or size. It’s a dumb question.

daveS, you may have a point, although I think it’s a tenuous one. To remove any ambiguity, I corrected my argument. I was thinking of a number used to express a size or quantity. It cannot be 1/2 or -1 or any other non-natural number.

Mapou: It assumes that natural numbers can be fully counted to determine a quantity or size. It’s a dumb question.

So you’re saying asking if the natural numbers are finite is incoherent?

No. I’m saying that the idea or claim that there is a set of all natural numbers is incoherent. Your question assumes that an infinite set already exists, the very thing you are trying to prove. It’s a dumb question.

Mapou: I’m saying that the idea or claim that there is a set of natural numbers is incoherent.

So, there’s no set of rational numbers or real numbers, or prime numbers?

Only finite sets can exist. There is no set of ALL natural numbers (Note: I corrected my comment @665 to say “all natural numbers”). If you had an infinite set of ANYTHING, you would be able to show it to me. You can’t. Saying that you can imagine it in your head is not proof of existence.

Mapou: Your question assumes that an infinite set already exists, the very thing you are trying to prove. It’s a dumb question.

In set theory, the assumption is that if there is a n element, there can be an n+1 element. This implies an unbounded set. If you reject this, then you reject much of mathematics which has relied on induction since the Classical Greeks.

Mapou: There is no set of ALL natural numbers (Note: I corrected my comment @665 to say “all natural numbers”).

You can have set theory with or without the axiom of infinity. When talking about set theory, it normally includes this axiom, and there is no inconsistency in adopting this axiom, even if you don’t like it.

So, there’s no set of rational numbers or real numbers, or prime numbers?

The answer, of course, is sure there “is”, if you create a definition for the set. It comes down to coherency and contradiction with respect to pure mathematical definition, an actuality when it comes to the real world we live in.

The former involves Cantor and Godel et al and is admittedly very interesting fodder for mental masturbation. (I would put the discussion of the nature of such squarely in the realm of philosophy, which many think is “dead.” Apparently not. Although there does seem to be real world implications if we just take it as tricky symbol manipulation and not worry about “what it really means.”)

The latter pertains to instantiation of any infinite set which seems to me to be impossible nonsense in the real world which I think boils down to a tricky manipulation of symbols.

Intuitively, there seems to be some sort of “Platonic reality” to infinity (of the various kinds) and numbers in general. Sir Penrose thinks our intuition is touching upon something transcendent and real when it comes to the subject of numbers and has spilt a lot of ink saying why he thinks so. Maybe he’s right.

Mapou: Your question assumes that an infinite set already exists, the very thing you are trying to prove. It’s a dumb question.

In set theory, the assumption is that if there is a n element, there can be an n+1 element. This implies an unbounded set. If you reject this, then you reject much of mathematics which has relied on induction since the Classical Greeks.

I don’t reject the concept of an expanding set at all. I have no problem with continually adding a new member to a set. But regardless of how many new members you add, the set is always finite.

Mapou: There is no set of ALL natural numbers (Note: I corrected my comment @665 to say “all natural numbers”).

You can have set theory with or without the axiom of infinity. When talking about set theory, it normally includes this axiom, and there is no inconsistency in adopting this axiom, even if you don’t like it.

I really have no problem with the axiom. I have a problem with the use of the word “infinity” in its name. It’s a misleading and self-contradictory name, IMO. In fact, I may even agree with most of Cantor’s ideas on expanding sets. I just disagree that he was working with infinite sets or that it is possible to work with infinite sets.

Mapou,
I don’t reject the concept of an expanding set at all. I have no problem with continually adding a new member to a set.

In your view, is it legitimate to regard 72,944,303,471,110 as a natural number? After all, you’ve never seen anyone construct that number, starting from 1, by repeatedly adding 1.

keith, sorry but I don’t see what your question @673 has to do with infinity or infinite sets. If you need a definition for natural numbers, Google is your friend.

Zachriel: So, there’s no set of rational numbers or real numbers, or prime numbers?

mike1962: The answer, of course, is sure there “is”, if you create a definition for the set.

That’s what ZFC set theory does with the axiom schema of replacement specification.

mike1962: The former involves Cantor and Godel et al and is admittedly very interesting fodder for mental masturbation.

As pointed out above, infinity is important in physics. See Newton, Mathematical Principles of Natural Philosophy, Philosophical Transactions of the Royal Society 1687.

Mapou: But regardless of how many new members you add, the set is always finite.

The set of natural numbers is unbounded.

Mapou: I really have no problem with the axiom. I have a problem with the use of the word “infinity” in its name.

keiths: In your view, is it legitimate to regard 72,944,303,471,110 as a natural number? After all, you’ve never seen anyone construct that number, starting from 1, by repeatedly adding 1.

If I might dive in, personally I think 72,944,303,471,110 is natural number. I can write a computer program to count up to it and display it on my KayPro (okay, maybe I would need a faster computer than that to do it), which is proof enough for me that it is a value with meaning in the real world. I cannot do that with all (or even “most” of) the values in a transfinite set.

On a slightly more serious note, 72,944,303,471,110 is obviously natural because it’s quantified and obviously so. No reason to even count up to it. It’s already there! 🙂

Zachriel: As pointed out above, infinity is important in physics. See Newton, Mathematical Principles of Natural Philosophy, Philosophical Transactions of the Royal Society 1687.

Oh, I most definitely agree, and already acknowledged that. There is symbolic manipulation thereof that has led to useful, real world results in physics. However, the results themselves do not contain infinities. Infinities must be cancelled out. They must be tossed out on their ears by the time any usefulness of the exercise is seen, either in terms of our understanding of the world, or our ability to make better toasters.

It is the nature of what is actually being manipulated which is the controversial thing in my thinking. Are infinities that are represented by the symbols mere mathematical tricks? Or do they correspond to something that our intuition is picking up on that is profoundly (and by necessity, if true, transcendentally) real? As I said, Sir Penrose thinks the latter and has spilt a lot of ink explaining why he thinks so.

My best friend has a PhD from Harvard University math department with a specialty in topology. (Weird stuff. And mostly useless with respect to practical reality. However, he knows his maths.) We have spent quite a number of hours discussing infinity, and I think I understand the things he tries to explain to me. I’ve also read a lot of books that deal with the subject. In the end, for me…

I think it’s practical nonsense, symbolic trickery, with perhaps some sort of correspondence to something else that may be real that transcends the physical world. After all, 1 + 1 = 2 seems to transcend any physical instantiation, so maybe transfinite sets do too. But I have yet to see an instantiation of a transfinite set. Nor do I see how it could even be possible in theory.

keith, sorry but I don’t see what your question @673 has to do with infinity or infinite sets.

It’s simple.

You write:

I don’t reject the concept of an expanding set at all. I have no problem with continually adding a new member to a set. But regardless of how many new members you add, the set is always finite.

If you accept the existence of large and physically unrealizable finite quantities such as a googolplex, what is your principled reason for rejecting infinity?

Zachriel @678, I don’t care how you call a set or series that can expand. It is finite. I’m getting tired of your vapid replies that add nothing to your arguments for the existence of infinite sets or infinity, arguments that I have already refuted.

If you accept the existence of large and physically unrealizable finite quantities such as a googolplex, what is your principled reason for rejecting infinity?

All finite quantities can be written down (or represented) in a sufficiently large finite universe. Infinity cannot be written down in any finite universe. Any number yields another number if you as 1 to it. That is, any number except infinity. Why is infinity the exception? Answer: it does not exist. Live with it.

Now, I may be stating the obvious, but it seems to me that an instantiation of a transfinite set as an actual ontology would necessarily require trans-temporality in an unbounded reality or as an unbounded reality.

mike1962: Infinities must be cancelled out. They must be tossed out on their ears by the time any usefulness of the exercise is seen, either in terms of our understanding of the world, or our ability to make better toasters.

No, they’re not tossed out on their ears, but essential steps to finding the answers.

mike1962: Are infinities that are represented by the symbols mere mathematical tricks? Or do they correspond to something that our intuition is picking up on that is profoundly (and by necessity, if true, transcendentally) real?

In the calculus, they represent dividing the continuum into smaller and smaller pieces, to the limit.

mike1962: After all, 1 + 1 = 2 seems to transcend any physical instantiation, so maybe transfinite sets do too.

Yes, mathematics is an abstraction, and is a model when it conforms in one way or another to the world. But all models are incomplete and imperfect. Just because zero causes some people philosophical discomfort doesn’t change this. People get use to it, like the Earth moving, or negative numbers.

Mapou: I don’t care how you call a set or series that can expand.

Unbounded sets are normally called infinite.

Mapou: arguments that I have already refuted.

Actually, your argument was that you didn’t like using the term “infinity”. There are many problems in mathematics and physics that require the use of unbounded limits, by whatever name you call them.

Zachriel: No, they’re not tossed out on their ears, but essential steps to finding the answers.

Having re-read my words, I think I was not clear. They have to be cancelled out by the time the equations have any final value. The time-worn issue of integrating General Relativity and Quantum Physics is an example.

In the calculus, they represent dividing the continuum into smaller and smaller pieces, to the limit.

But that’s just restating the “problem” using different words. Symbols representing “the continuum” are manipulated. But at the conclusion, the infinities are out of the picture.

I never said the manipulation of symbols that respresent infinities were not useful. Of course they are, else nobody would be bothering to discuss them here.

Yes, mathematics is an abstraction, and is a model when it conforms in one way or another to the world. But all models are incomplete and imperfect. Just because zero causes some people philosophical discomfort doesn’t change this.

I would have to agree.

People get use to it, like the Earth moving, or negative numbers.

I don’t think I would put those at the same level of abstraction. Neither of those require trans-finite space and trans-temporality to conceivably have an instantiation. Some ideas are more abstract than others. Infinities are the ultimate abstractions.

If the universe is finite, then some finite quantities are too large to be represented within it. Do those quantities exist, according to you?

If so, then why not infinity?

No. Nothing exists unless it does. It’s that simple. If you can show me your infinite set (or any set, for that matter), it exists. Otherwise, it’s a figment of your imagination. Adding ‘…’ or ‘etc.’ at the end of your series does not count.

mike1962: I don’t think I would put those at the same level of abstraction. Neither of those require trans-finite space and trans-temporality to conceivably have an instantiation.

But that’s just restating the “problem” using different words. Symbols representing “negative sheep” are manipulated. But at the conclusion, there’s no negative sheep.

Zachriel: But that’s just restating the “problem” using different words. Symbols representing “negative sheep” are manipulated. But at the conclusion, there’s no negative sheep.

Are you saying you don’t think there’s a difference in the level of abstraction with regards to infinities and negative numbers?

mike1962: Are you saying you don’t think there’s a difference in the level of abstraction with regards to infinities and negative numbers?

If by abstraction, you mean there are no negative sheep, then yes. The reaction is analogous to when negative numbers were introduced.

Negative numbers “… darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple” .

mike1962: Are you saying you don’t think there’s a difference in the level of abstraction with regards to infinities and negative numbers?

Zachriel: If by abstraction, you mean there are no negative sheep, then yes [there is no difference].

No, I was referring to the level of abstraction (you do think there are levels of abstraction, don’t you?), and what you think about it, not reactions from some dude in the past.

All finite quantities can be written down (or represented) in a sufficiently large finite universe. Infinity cannot be written down in any finite universe. Any number yields another number if you as 1 to it. That is, any number except infinity. Why is infinity the exception? Answer: it does not exist. Live with it.

If you prefer to work in this very austere mathematical environment, that’s fine, of course.

You stated earlier that physicists always end up using discrete methods, despite the fact that they sometimes claim otherwise. Do you think it matters whether physicists specifically accept the “existence” of infinity? In the end, will there be any difference?

mike1962: No, I was referring to the level of abstraction (you do think there are levels of abstraction, don’t you?), and what you think about it, not reactions from some dude in the past.

Abstractions are abstractions. Negative sheep don’t exist. Zero sheep don’t exist.

mike1962: You stated earlier that physicists always end up using discrete methods, despite the fact that they sometimes claim otherwise.

Physicists often work with infinities. See Newton 1687.

All finite quantities can be written down (or represented) in a sufficiently large finite universe. Infinity cannot be written down in any finite universe. Any number yields another number if you as 1 to it. That is, any number except infinity. Why is infinity the exception? Answer: it does not exist. Live with it.

If you prefer to work in this very austere mathematical environment, that’s fine, of course.

What is austere about it? I use the same calculus that everybody else uses. I see no infinity in it though.

You stated earlier that physicists always end up using discrete methods, despite the fact that they sometimes claim otherwise. Do you think it matters whether physicists specifically accept the “existence” of infinity? In the end, will there be any difference?

No, of course. Physicists can claim anything they want. In the end they die like everybody else but the truth remains.

Okay. Then by your logic, there must be a largest finite number. What is it, and what happens if you add 1 to it?

Any number that you add 1 to becomes a new number. You can’t do that to infinity because it does not exist. If you can represent a number in some manner, the representation exists. I don’t know what the largest number that has ever been represented is but, you know what? I don’t really care. Your point is what again?

You stated earlier that physicists always end up using discrete methods, despite the fact that they sometimes claim otherwise. Do you think it matters whether physicists specifically accept the “existence” of infinity? In the end, will there be any difference?

I just now realized that you asked me this question before but in different words. I should say that it makes no difference as far as the non-existence of infinity is concerned. But it makes a HUGE difference in the way we understand and look at reality once we realize that reality is discrete and finite. I already answered this question @255 above.

What is austere about it? I use the same calculus that everybody else uses. I see no infinity in it though.

You’ve never taken an integral from something to infinity?

Taylor series are sums from one to infinity. Einstein used a Taylor series in his derivations.

The area under a normal curve is one if you integrate from minus infinity to infinity.

Gravitational and electrical and magnetic fields extend out to infinity. There is nothing in their formulas which says they end some place.

I’m finding this conversation a bit weird. Limits, asymptotic behaviour, you can’t work with these things unless you can deal with the infinity large and the infinitely small. How do you know if an infinite series converges or diverges? The famous one, Zeno’s paradox, is a classic example:

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + . . . = 1

How do you establish that? You give me an tolerance, a distance from the answer 1 that you want me to get to and then I tell you how many terms of the sequence I need to add up to get to that tolerance. AND I can do that no matter how small you make that tolerance.

The basic concept of the limit underlies ALL of calculus since a derivative is defined by a limit.

Sigh.

Never mind. A lot of good work is and can be done just by utilising techniques developed by the theorists. I know that. But the theories have to be solid and many of them sit on the infinitely large and the infinitely small. And, guess what, it works.

Zachriel: Abstractions are abstractions. Negative sheep don’t exist. Zero sheep don’t exist.

I agree that negative sheep don’t exist. But I do think that zero sheep can exist. Class of “sheep.” Instantiation of class of “sheep.” Zero instantiation of class of “sheep.” But I cannot imagine a negative instantiation of the class “sheep”, can you?

Anyway, do you think these are on the same level of abstraction as infinities, and infinities of infinities? If so, we’ll have to simply disagree.

You stated earlier that physicists always end up using discrete methods, despite the fact that they sometimes claim otherwise.

No, I didn’t. Not at all.

Physicists often work with infinities. See Newton 1687.

Agreed. (For the third time.)

I don’t think you’re carefully reading what I write.

You can jump up and down and foam at the mouth as much as you want but I see no infinity in your argument. Nothing ever reaches infinity. Keep jumping.

Jared: Never mind. A lot of good work is and can be done just by utilising techniques developed by the theorists. I know that. But the theories have to be solid and many of them sit on the infinitely large and the infinitely small. And, guess what, it works.

However, these “infinities” that theories “sit on” are derivations of definitions of transfinite sets. Symbolic definitions. Nobody ever actually deals with any instantiated infinities. No matter how you slice it, it boils down to manipulation of symbols.

They are amazing. And they work. But why do they work?

The “controversy”, for me, and some others, is whether or not infinities have a real, you might say, “platonic” reality or ontology, regardless of the symbolic definitions humans have devised.

Stuff like that.

I believe that Mapou is objecting to an instantiation of any infinite class or set in the Real World that we inhabit, which, of course, I think he would be correct, if the Real World is limited, and most physicists think it is.

You can jump up and down and foam at the mouth as much as you want but I see no infinity in your argument. Nothing ever reaches infinity. Keep jumping.

I know you disagree with me but it would be nice if you were a little less rude. Foaming at the mouth? I know moderation is practically non-existent on UD at the moment but . . . please.

You have seen and done definite integrals out to infinity though? You have had to deal with the concept. You must have done. Did you just learn what hoops to jump through and bite your tongue for years of higher education?

mike1962 #707

However, these “infinities” that theories “sit on” are derivations of definitions of transfinite sets. Symbolic definitions. Nobody ever actually deals with any infinities. No matter how you slice it, it boils down to manipulation of symbols.

Well, how did Cantor show that the size of the set of real numbers is larger than the set of integers if he didn’t deal with infinities?

What about the hyperreal numbers?

And the surreal numbers?

And the imaginary numbers?

The irrational numbers have infinite, non-repeating decimal expansions. The Greeks figured out something was weird about sqrt(2). Same with Pi. And e.

Have you read about Cantor’s continuum hypothesis?

Sure, they “work”, but why do they work?

Because they are sitting on a firm theoretical foundation which transcends the material world?

The controversy, for me, and others like me, is whether or not infinities have a real, you might say, “platonic” reality or ontology, regardless of the symbolic definitions humans have devised.

Well, using them works and gives good, solid, practical, useable results. And we’ve now got a solid underpinning for how to work with them.

I believe that Mapou is objecting to an instantiation of any infinite class or set in the Real World that we inhabit, which, of course, I think he would be correct, if the Real World is limited, and most physicists think it is.

But surely he would agree that’s there’s more to existence than the physical, ‘real’ world. He sounds like the real materialist!! Seriously, I thought those with a more theological outlook than myself would embrace and welcome discussions of the infinite. But I do find this whole conversation confusing regarding what some people are saying. I should probably just shut up.

I tell what I really don’t get is the hostility to some of the ideas I’ve been trying to explain.

Okay, maybe you don’t understand what I’m talking about or like it or care or think it’s important. But why the hostility from people like Mapou and phoodoo? Why the association with materialism?

(In fact, with his insistence on solid, real world, measurable quantities Mapou sounds like the real materialist.)

If you really want to have a dialogue and you want to discuss ideas but you treat some with scorn, derision and outright hostility then you can’t really expect people you disagree with to want to talk to you. I’m beginning to wonder why I bothered. ‘Foaming at the mouth’ Sheesh.

I know you disagree with me but it would be nice if you were a little less rude. Foaming at the mouth? I know moderation is practically non-existent on UD at the moment but . . . please.

I refuse to be polite to you, Jerad, because I have seen your dishonesty in this thread. It is insulting and offensive to me.

You have seen and done definite integrals out to infinity though? You have had to deal with the concept. You must have done. Did you just learn what hoops to jump through and bite your tongue for years of higher education?

mike1962: I agree that negative sheep don’t exist.

Okay.

mike1962: But I do think that zero sheep can exist.

Zero sheep is indistinguishable from zero marbles.

mike1962: Anyway, do you think these are on the same level of abstraction as infinities, and infinities of infinities?

The way abstraction has been used on this thread is to refer to mental constructs. Negative numbers, zero, and infinity are mental constructs, that is, abstractions with no material reality.

As we pointed out, people once had troubles with negative numbers, then decided to use them but pretend they were just computational devices, until today, they seem just as normal as natural numbers.

mike1962: No, I didn’t. Not at all.

Sorry, misattribution.

mike1962: Are you a platonist with respect to numbers?

If wishes were fishes, the oceans would be riches.

mike1962: No matter how you slice it, it boils down to manipulation of symbols.

What is austere about it? I use the same calculus that everybody else uses. I see no infinity in it though.

Well, with no infinite sets, it seems to me your system would lack many of the standard tools of mathematics. For example, the Hilbert spaces I referred to above, whether separable or not, make no sense if infinite sets do not exist. They are complete inner product spaces over R or C by definition, so the underlying set of vectors must be uncountable. As a specific example, l^2 (little L 2).

Maybe you can kludge together some sort of discrete approximation to things such as these, but it seems like a lot of work if in the end, you get approximately the same answer.

Are there _any_ existing theories in physics that _don’t_ involve so-called “infinite” sets? Like I said, I know nothing about physics, so maybe there are; if so, I’d like to know about them.

I was also wondering how you would deal with sequences. For example, the sequence of partial sums of the harmonic series. How would you prove that it diverges? Or do you believe that it diverges in the first place?

I tell what I really don’t get is the hostility to some of the ideas I’ve been trying to explain.

Okay, maybe you don’t understand what I’m talking about or like it or care or think it’s important. But why the hostility from people like Mapou

The only rational explanation seems to be that he has not yet made the transition to adulthood. His illogical and insulting posts do not seem to be the product of a socially mature adult mind. This behavior is not confined to this thread; it pervades all his posts here at UD.

You are wasting your time trying to have a logical and civil dialog with him. He is neither logical nor civil.

Let’s try this: have a dialog with me instead, and totally ignore anything that Mapou posts. Let’s see where it goes.

Jerad qutoing me: Nobody ever actually deals with any infinities.

Actually, you managed to quote my post before I edited it. But I cannot say that I object to anything you said in your reply.

I should probably just shut up.

Nawwww. What fun would that be?

Not sure why Mapou gets worked up about the subject.

I am admittedly quite disposed to the platonic (small P) side of the fence with respect to all of this. I would have to say I agree with Mapou in certain ways with regards to instantiation of transfinite classes and sets. But my intuition tells me the classes and sets have an platonic ontological reality beyond mere symbol crunching, as non-commonsensical as it may seem to common sense.

I heard that Cantor claimed that God revealed his transfinite ideas to him. Does anyone have a source for this?

daveS @715, I have Mathematica on my computer and I can use it to do all sorts of beautiful things that supposedly assume the existence of infinity. And yet, I can assure you that my computer is as discrete and finite as can be. Like I said earlier, I use Fourier analysis to decompose waveform data into a frequency spectrum. Guess what? I feed it discrete values and I get discrete values in return. No infinity and no infinitesimals ever enter the picture.

How is that possible? The reason is that it is a lie that infinity is ever used in any of the calculations. And it is not just a small lie or a simple misunderstanding. It is a big lie, a HUGE LIE.

Mapou: And yet, I can assure you that my computer is as discrete and finite as can be.

Sure. That’s what we mean by a set, treating a collection as a single entity. So the set of mammals, the set of cat’s eye marbles, the set of natural numbers. We put them between curly brackets and then discuss them as a single entity.

It’s in the nature of humans to divvy up the universe. Set theory is how we add mathematical rigor to these divisions.

Mapou: And yet, I can assure you that my computer is as discrete and finite as can be.

Sure. That’s what we mean by a set, treating a collection as a single entity. So the set of mammals, the set of cat’s eye marbles, the set of natural numbers. We put them between curly brackets and then discuss them as a single entity.

It’s in the nature of humans to divvy up the universe. Set theory is how we add mathematical rigor to these divisions.

I was waiting for something of substance. Silly me. How does that prove that infinity is required to do ANY kind of math when I just proved the opposite?

daveS @715, I have Mathematica on my computer and I can use it to do all sorts of beautiful things that supposedly assume the existence of infinity. And yet, I can assure you that my computer is as discrete and finite as can be. Like I said earlier, I use Fourier analysis to decompose waveform data into a frequency spectrum. Guess what? I feed it discrete values and I get discrete values in return. No infinity and no infinitesimals ever enter the picture.

How is that possible? The reason is that it is a lie that infinity is ever used in any of the calculations. And it is not just a small lie or a simple misunderstanding. It is a big lie, a HUGE LIE.

It’s been a while since I used Mathematica, but I agree it’s an amazing piece of software.

Can you give some specific examples of these calculations that supposedly assume the existence of infinity?

I do understand that you can compute Fourier coefficients (approximately at least and sometimes exactly) without this assumption.

Mapou: How does that prove that infinity is required to do ANY kind of math when I just proved the opposite?

We simply responded to your suggestion that computers can’t work with the mathematics of infinity. They can — the same way people do — by defining a set to contain a non-finite assemblage, such as the natural numbers.

As for most computer integration, that’s usually done numerically which provides for approximate answers. However, it’s possible to solve some integrations using the calculus of the infinitesimal, something Newton used to great effect in Principia.

I think the one thing this thread very clearly does demonstrate is that there is a lot of stuff taught in schools, even at the best universities in the world, that is without a doubt just mental nonsense.

When you start taking mental concepts such as a “beginning”, or an “end”, or “unlimited”, or “all” or “large” , and then you pretend that you can make this into a definitive math problem, you are already just playing a game. There is no number for large, or a number for middle, or a number for vast or a number for time…these are things we imagine in our brains.

So when people start talking about how Cantor “proved” there are some infinities bigger than others, and this is taught as a reality in major universities, people have a right to be offended. It is an insult to the whole concept of knowledge. It would be like teaching at a university that red is beautiful, we have proved it. Or like teaching that time is frisbee shaped in heaven, we have proved it. And then middle school kids go around telling their friends that their brother taught them time was frisbee shaped. It can be proven mathematically.

No one can prove nonsense wrong, anymore than one can prove it is right. If someone was reading this thread hoping to see some insight as to why the mathematicians believe it is reasonable to side with Cantor, they will be sorely disappointed. The mathematicians are not using logic, or reason, or facts, they are just saying it is so, and hoping no one calls them on it. NO realities in life are solved by this math problem. NO new information about the world is gained by this absurd use of definitions. It makes a complete mockery of the concept of knowledge.

Once we see that you can make such absurd statements, such as those by Cantor, and then have it turned into a commonly taught believe, then everything else we are taught by supposedly trained professionals deserves to be second guessed-and at times ridiculed even. I think it undermines the whole profession of teaching. Cantors theory doesn’t teach us how to build a bridge stronger, or a better airplane, or how to find alternate energy sources. It teaches nothing. No platform of knowledge is made stronger by this imaginary card game.

This thread shows that a lot of things are just mental jokes like an emperor with no clothes that no one wants to mention.

Mapou: How does that prove that infinity is required to do ANY kind of math when I just proved the opposite?

We simply responded to your suggestion that computers can’t work with the mathematics of infinity. They can…

Funny since I just finished showing the opposite. There is no mathematics of infinity, otherwise my computer could not work with it. How many brains are contained in this “we” that you keep referring to? Are you a fruitcake or something? Or several fruitcakes?

The mathematicians are not using logic, or reason, or facts, they are just saying it is so, and hoping no one calls them on it.

The irony is that despite your rhetoric, it is the mathematicians who are being rigorous, while you, Mapou and Joe are succumbing to emotion and irrationality.

Cantor could teach you something about disciplined thinking if you would only listen.

During his university studies Cantor felt a deep calling from God to study philosophy and mathematics, rather than more lucrative pursuits. His faith sustained him during long years of rejection when the mathematical establishment dismissed his concept of the transfinite. When weaker men would have abandoned their work, Cantor perseve

Of note: I hold ‘growing large without measure’ to be a lesser quality infinity than the infinity in which ‘a fraction in which the denominator goes to zero’. The main principle for why I hold growing large without measure to be a ‘lesser quality infinity’ is stated at the 4:30 minute mark of the following video:

It is also interesting to note that the conflict of reconciling General Relativity and Quantum Mechanics arises from the inability of either theory to successfully deal with the Zero/Infinity conflict that crops up in different places of each theory:

Moreover there is actual physical evidence that lends strong support to the position that the ‘Zero/Infinity conflict’, we find between General Relativity and Quantum Mechanics, was successfully dealt with by Christ:

Moreover, as would be expected if General Relativity, Quantum Mechanics/Special Relativity (QED) were truly unified in the resurrection of Christ from death, the image on the shroud is found to be formed by a quantum process. The image was not formed by a ‘classical’ process:

As well, as would be expected in such a ‘singularity’ reconciling Gravity with Quantum Mechanics, Gravity appears to have been overcome in the resurrection event of Christ:

Verses and Music:

Supplemental note:

Get Joe to watch this one (from BA77’s post), he doesn’t believe in Cantor’s mathematics.

Georg Cantor – The Mathematics Of Infinity – video

https://vimeo.com/96082227

0 is the sum of all positive and negative numbers. It is unique in that it is neither positive nor negative. 0 is the loneliest number. This is why everything comes from nothing and amounts to nothing. It is the reason that we live in a yin-yang or symmetric universe. The ultimate law of physics is the conservation of nothing. Motion is caused by the universe correcting a violation to the conservation of nothing.

0 is why the universe is ONE. 🙂

To conclude, what did the Zen master say to the hot dog vendor? Answer: Make me one with everything. 😀

Actually the fact is I have proven that his one-to-one correspondence is contrived,rather than derived, with respect to infinite sets in which one set is a proper subset of the other. That means his one-to-one correspondence is not what it appears as obviously there isn’t one and only one match between the two sets.

Leave it to Jerad to not be able to grasp that simple fact.

Infinity is crackpottery for a surprisingly simple reason.

Compared to infinity any finite quantity is infinitely small.That’s it. In other words, if one accepts infinity, one must accept a reality where quantities are both finite and infinitely small at the same time. Alternatively, infinity cannot exist because it cannot be compared to any finite quantity without introducing a logical contradiction.

It gets worse, much worse. Without infinity, continuous structures, which are infinitely smooth by definition, cannot exist. Not long before his death, Einstein wrote to his friend Besso:

In other words, physics is sitting on a mountain of crap.

Mapou:Compared to infinity any finite quantity is infinitely small. That’s it. In other words, if one accepts infinity, one must accept a reality where quantities are both finite and infinitely small at the same time.Yikes! No!

Your first statement concerns the ratio of a finite quantity to infinity. Your second statement conflates the finite quantity (which is finite) with the ratio (which is an infinitesimal).

alter-M:

Compared to 1000, 1 is only 1/1000th. That’s it. In other words, if one accepts 1000, one must accept a reality where quantities are both 1 and 1/1000th at the same time.Zacky:

Is this reaction supposed to intimidate me or what? Let’s take a look at your logic, if we can call it that.

I do no such thing.

1/1000 is not a comparison; it’s just a number. A comparison is not a number. It is a test that returns a Boolean value, either true or false. In other words, I am asking two questions:

1. Is 1 infinitely smaller than infinity, yes or no?2. Is 1 finite, yes or no?

If both answers are affirmative, then there is a contradiction.

PS. I realize that Darwinists and materialists are weavers of lies and deception but you take the cake, Zacky. You are stupider than I thought. But since you love to refer to yourself with the plural “we”, it follows that you are all stupid.

If you believe in infinity, you are a crackpot. Sorry.

Zero is not false.

Mapou:1. Is 1 infinitely smaller than infinity, yes or no?That question concerns a ratio. The ratio is an infinitesimal.

“Infinity is crackpottery for a surprisingly simple reason.”

Everything from nothing is crackpottery.

Zacky:

So what?

Mapou:So what?So now we have the following answers.

1. The ratio of 1 to infinity is an infinitesimal.

2. 1 is finite.

There is no contradiction.

Zacky:

Amazing. Of course there is a contradiction and it is a blatant one. Does it not follow that, if the ratio of X to infinity is an infinitesimal, that X is infinitely smaller than infinity? And is it not true that infinity is infinitely greater than any finite value X? If true on both counts, then infinity is crackpottery.

Mung:

Says the man who insists that plural = singular.

Don’t feel too bad, Zacky. Most Christians believe in infinity just as you do. 😀

Mapou:Does it not follow that, if the ratio of X to infinity is an infinitesimal, that X is infinitely smaller than infinity?Yes, if the ratio of X to infinity is an infinitesimal then the ratio of X to infinity is an infinitesimal.

You don’t get it, Zachriel. When you compare two quantities, X and Y, you are not asking for a ratio. You are looking for a truth value. Heck, forget ratios. This is not the way comparisons are done in the first place. A CPU does not use ratios to compare two values. It just subtracts one value from the other. The result can be negative, positive or zero. A CPU cannot even compare infinity to another number because its registers are finite. This alone refutes infinity. We know infinity is greater than a finite number, not because we can perform the comparison operation but because it is true by definition.

So your entire ratio argument is bogus right off the bat.

Mapou:A CPU cannot even compare infinity to another number because its registers are finite.Computers can be programmed for the mathematics of infinity, just like they can for standard arithmetic, or non-Euclidean geometry, or whatever.

Mapou:When you compare two quantities, X and Y, you are not asking for a ratio. You are looking for a truth value.You need to be specific. Do you mean X is less than Y. Sure. But you wanted a comparative, that is, how much smaller.

Mapou:It just subtracts one value from the other. The result can be negative, positive or zero.So you don’t want a truth value, nor the ratio, but the difference. That’s fine. The difference is infinity.

Mapou:1. Is 1 infinitely smaller than infinity, yes or no?2. Is 1 finite, yes or no?

1. The difference of 1 and infinity is infinity.

2. 1 is finite.

Still no contradiction. By the way, and there are special rules for the mathematics of infinity.

You need to be careful with infinity, because infinity is not a number. It’s a mathematical concept.

Zachriel:

Nonsense. There are three possible comparison operations that can be performed on X and Y: greater than, less than and equal. Each returns a Boolean value, not a number. Wake up.

Mapou:There are three possible comparison operations that can be performed on X and Y: greater than, less than and equal.Then 1 < infinity.

1 is finite.

Done.

Bzzzt. Nobody is saying that 1 is not finite. 1 is not just smaller than infinity. It is

infinitelysmaller than infinity while being finite at the same time. You cannot say this about any other comparison. That’s the qualitative difference that you insist on ignoring.Mapou:1 is not just smaller than infinity. It is infinitely smaller than infinity while being finite at the same time.And that refers to either a difference or a ratio. Unless you mean some other comparison.

.

Give a rigorous mathematical definition of what

youmean by the phrase “infinitely smaller”.Then we can talk.

.

1. Since no machine in existence can subtract any number from infinity, infinity is bogus.

2. Since a finite number is infinitely smaller than infinity, infinity is bogus.

I rest my case.

Cantor, do you need a rigorous mathematical definition in order to understand that infinity is infinitely greater than any finite number?

And do you need a rigorous mathematical definition to understand that the inverse (any finite number is infinitely smaller than infinity) is also true?

If you do, I feel sorry for you.

Mapou:do you need a rigorous mathematical definition in order to understand that infinity is infinitely greater than any finite number?It’s your claim that you can provide a mathematical proof that “infinity is crackpottery”, even though mathematicians nearly universally accept the concept. That means it’s your responsibility to provide unambiguous definitions of your terms.

.

Well said.

.

Mapou,

I agree completely but I’m not sure why you insist on stating that most Christians believe in infinity.

William Lane Craig (who many consider a leading Christian spokesman) to my knowledge shares your opinion:

https://www.youtube.com/watch?v=4X6XKKGo5GY

.

If you want your claim to be taken seriously you need to define your terms.

.

Don’t be sorry. Cheer up and define your terms. Then we can talk.

.

Zacky:

Mathematicians can kiss my asteroid because they have a lame pony in this race. They’re already on the record for claiming that infinity exists and they will look bad if the opposite is shown to be true. Mathematicians are political/religious animals just like Darwinists, therefore they are not to be trusted.

My peers are the public. If a concept cannot be explained in simple terms that the average intelligent layperson can understand, it’s crap, IMO.

computerist:

Thanks for the link. Actually, most Christians, Catholics and Protestants, believe that God is infinitely knowledgeable and powerful. They call it omniscience and omnipotence. It’s heresy, IMO, the work of the devil 😀 . I say this because it introduces all sorts of logical and ethical problems.

Zero is an odd number.

Unlike all other numbers:

It is the only number that is neither positive nor negative. It is the only number, when divided by itself does not equal one.

It is the only divisor that produces an undefined result.

Repeated divisions by two still produce an even number.

It is unique. It is odd.

Mapou’s infinity argument reminds me of the following:

1) Nothing is better than complete happiness in life.

2) A ham sandwich is better than nothing

3) Therefore a ham sandwich is better than complete happiness in life.

Both arguments make a lot of sense… if you don’t think about them.

There is an even simpler proof that infinity is bogus: If a quantity cannot be counted by any possible computer, that quantity cannot exist.

I place the concept of infinity in the same category as Darwinism: a gigantic fraud perpetrated on the public for nefarious reasons. Some evil forces are trying extremely hard to keep humanity from acquiring certain forbidden knowledge about the universe. In particular, we are being kept from understanding the following:

1 The universe is necessarily discrete and absolute, and had a beginning.

2. Space (distance) is but a perceptual illusion.

3. A time dimension is pure crackpottery.

cantor,

With a name like cantor, you are wearing your religion on your sleeve, aren’t you? You expose your bias for all to see. How is that working for you?

.

I seem to have grossly overestimated Mapou. His latest 2 posts read like something a tween might write.

He could learn a lot here on UD if he would constructively engage with those more educated than he.

Instead he posts inscrutable gobbledygook.

.

I’m glad you appreciate my ideas, cantor. It fills me with great warmth. Here’s a little question that’s bothering me. Maybe you can help. I mean, an accomplished and brilliant mathematician like you should have no trouble knowing the right answer. A simple yes or no will suffice. I assure you it is not a trick.

Can a body move in spacetime?

PS. Zachriel is also welcome to take a shot at it.

Mapou @ 39

Whether you believe in infinity or not, don’t tell me computers can’t “count” infinity. Every computer can calculate infinity. Heard of Basel problem ? Try calculating the sum. Your computer will do it in a jiffy. The answer is

`Pi^2/6`

My, there is much ado about nothing!

Me_Think:

That’s news to me and I’ve been around computers for a long time.

Every computer can calculate infinity…. Lol!!!!

Materialists say the darnest things……….

Wonder how my 486 SX did it?

Mapou – if infinity doesn’t exist, then what’s the largest prime number? Or, perhaps easier, how do we identify it?

Bob,

You do realize that numbers aren’t actually real things, right?

What do you think the largest prime number is? Since numbers are just concepts, there is no largest number. Infinity is nether the largest nor the smallest number, since it doesn’t represent anything.

Mapou:Mathematicians can kiss my asteroid because they have a lame pony in this race. They’re already on the record for claiming that infinity exists and they will look bad if the opposite is shown to be true.Infinity is an abstraction, just like -1 or a hyperbolic paraboloid. Your claim is that the concept is inconsistent. To show that requires a mathematical proof.

Mapou:My peers are the public. If a concept cannot be explained in simple terms that the average intelligent layperson can understand, it’s crap, IMO.Like quantum mechanics or general relativity, or for that matter, the circuits in your CPU.

dgw:Zero is an odd number.Heh.

cantor:Both arguments make a lot of sense… if you don’t think about them.Ha!

Mapou:There is an even simpler proof that infinity is bogus: If a quantity cannot be counted by any possible computer, that quantity cannot exist.Argument by redefinition! Tee hee!

ETA: By that definition, there is no decimal expansion of 1/3.

phoodoo:You do realize that numbers aren’t actually real things, right? What do you think the largest prime number is? Since numbers are just concepts, there is no largest number. Infinity is nether the largest nor the smallest number, since it doesn’t represent anything.So close! You do realize that infinity is a concept, just like numbers are concepts?

alter-phoo: Infinity is nether the largest nor the smallest number, since it is not a number.

Bob O’H,

The largest prime number? What does that have to do with reality and the universe? You should ask instead, what is the largest

knownprime number?Infinity is for morons and idiots. No wonder materialists and Darwinists love it so much.

Mapou:What does that have to do with reality and the universe? You should ask instead, what is the largest known prime number?So there are finite of prime numbers determined by the largest yet found? Or can we prove they go on ad infinitum?

Zacky:

No, it requires simple logical proof that anybody can understand. I’m claiming that the concept has nothing to do with reality because it cannot be computed by any possible computer that we can think of. Therefore it’s crap.

There is no reason that the circuits of a CPU cannot be explained to a layperson in a language that they can understand. Don’t bring QM into the discussion since physicists have no clue what is going on either.

PS. I’m still waiting for you and your infinity buddy, cantor, to answer the question I posed earlier:

Can a body move in spacetime?This should not be so hard since you guys are so smart, right? After all, did not Einstein teach that bodies move along their worldlines or geodesics? Stop being a wussy.

Zacky:

All kinds of series can go on ad infinitum. So what?

Zacky:

How do you figure? It’s a logical proof that infinity does not exist in the universe. The universe is finite for this reason and it had a beginning.

No infinite series exists because it cannot be computed. Live with it.

Mapou:No, it requires simple logical proof that anybody can understand.A simple proof is preferred, however, any valid proof would be acceptable.

Mapou:I’m claiming that the concept has nothing to do with reality because it cannot be computed by any possible computer that we can think of.Actually, computers can be programmed to calculate infinities, just like they can be programmed to calculate using integers or imaginary numbers. They can’t

countto infinity, however, because they are finite machines, which, in case you missed it, means not infinite.Mapou:Don’t bring QM into the discussion since physicists have no clue what is going on either.So, according to the Mapou Principle, quantum mechanics is “crap” because it can’t be explained to an intelligent layperson.

Mapou @ 42 , Andre @ 43,

I repeat :

Mapou:It’s a logical proof that infinity does not exist in the universe.Infinity is an abstract concept, like -1 or a geometric line. All mathematics is abstraction.

Mapou:No infinite series exists because it cannot be computed.You seem to keep conflating countable with computable. In any case, in your medieval mathematics, there is no decimal expansion of 1/3. Not sure your regressive mathematics will catch on.

Mapou:All kinds of series can go on ad infinitum.You just used the word infinitum, meaning infinity.

Infinity is a concept, and in math infinite series has been computed. What do you think is Basel series?

I now fully realize that I’m arguing with willing morons.

Zacky @34:

You do realize that abstractions do not exist, right?

I have no problem with expanding 1/3. Maybe you do.

Wow. I can also talk about angels dancing on the head of a pin.

Give it up, Zacky. You got your foot in your mouth.

Mapou:You do realize that abstractions do not exist, right?That’s right. Abstractions don’t exist as physical realities. So 3+2i, a conic section, and -1, are abstractions, which may or may not represent some aspect of reality. Infinity is an abstraction just like the number 2.

http://www.youtube.com/watch?v=E-kMsBMh6Ng

Mapou:I have no problem with expanding 1/3.Nor do we! In decimal, it’s infinite and repeating, but not in ternary.

Mapou:Infinity is crackpotteryMapou:All kinds of series can go on ad infinitum.There you are.

I think Zacky suffers from some form of autism. I’ve had enough. Adios.

Infinity is the unattainable goal. It’s very real.

If zero is even, why can’t I divide by zero?

Maybe someone here has a computer that can divide by zero while it’s busy counting to infinity?

Speaking of crackpottery, belief that the left and right hemispheres of the brain represent yin and yang is crackpottery. So is the believe that everything came from nothing.

But once You believe one crackpot idea why not believe them all?

Perhaps infinity is a left brain concept and very real and not infinity is a right brain concept and very unreal.

Sigh.

I don’t understand why the majority of ID-supporting UD contributors are ignorant of the work done bey Georg Cantor in mathematics which is now an accepted part of mathematics.

There are different ‘sizes’ of infinity. The smallest one is aleph-naught, the ‘size’ of the natural counting numbers. Also referred to as countably infinite.

Any set which can be lined up, element per element, with the natural counting numbers is also said to be countably infinite. This lining up is called a one-to-one correspondence. IT DOES NOT refer to the value 1 but one element in one set matched with one and only one element in the second set.

Countably infinite sets include the natural counting numbers, the positive even integers, the positive odd numbers, the primes, the rational numbers (nice proof that), etc.

The real numbers are a ‘larger size’ of infinity. It is not possible to put them into a one-to-one correspondence with the natural counting numbers. And there’s a very nice proof of that!!

I hope this helps a little.

If we add an even number to an even number we get an even number. If we had an even number to an odd number we get an odd number. And if we add an odd number to an even number we get an odd number. And if we add an odd number to an odd number we get an even number.

Let X = any undisputed odd number

Let Y = any undisputed even number

0 + X = X

0 + Y = Y

0, even though odd, is even. 😎

Accepted but apparently not very useful. For example what difference would it be to say that a set of infinite numbers is larger than any of its also infinite proper subsets?

That is we keep the natural one-to-one correspondence used to determine one is a proper subset of the other and the cardinality is determined by a function that maps across positioning in the set.

Mapou

“I think Zacky suffers from some form of autism. I’ve had enough. Adios.”

LOL, A guy named Charles Martel from another forum which Zachy frequented as a time waster said that they figured he suffered from Asperger Syndrome. They banned him over there and someone said he came to this forum and sure enough, his M.O. hasn’t changed one iota.

Joe #64

Not very useful in a practical, everyday situation I grant you. In fact, practically useless. But, at the time, mathematics was in the throes of trying to make sure its foundations were rock solid. and there were certain concepts, like limits, that hadn’t been firmly established theoretically.

Cantor’s work was EXTREMELY controversial at the time. Many mathematicians refused to accept it. But now, it’s considered by most to be a building block of mathematics.

When you’re dealing with finite sets things are always easier. 🙂 Also, even with finite sets there are usually many ways to make a one-to-one correspondence between two sets.

Here’s an infinite set, let’s call it Z+ = {1, 2, 3, 4, 5 . . . .}

Here’s another infinite set E = {2, 4, 6, 8, 10 . . . . }

You can make the correspondence in this way:

Z+ –> E

1 –> 2

2 –> 4

3 –> 6

etc

This correspondence (or rule or function) is nice because we can even state a simple rule that describes it:

C: the rule that maps an element,n, of Z+ to 2n in E

But we could have made the following correspondence:

1 –> 4

2 –> 2

3 –> 8

4 –> 6

etc

We still get a one-to-one correspondence (i.e. every element of Z+ gets matched to one and only one element of £), it’s harder to define but no element in either set gets left behind.

Also, sometimes you deal with infinite sets that are not easy to ‘order’. For example, consider the rational numbers, Q, made up of all ratios of integers. So 1, 2, 3, 4 . . . are in there as are 1/2, 1/3, 1/4, 1/5 . . . and 2/3, 2/5, (2/4 or 1/2 already appears), 2/7 are in there as are 3/2, 3/4, 3/5 . . .

Finding a way to assign (or contrive if you like) a one-to-one correspondence between Q and Z+ is much tricker although it is possible. In fact, finding such a correspondence is part of one proof that the real numbers have a larger cardinality than the natural counting numbers, Z+.

Jerad:

The problem is not that infinity is not a viable mathematical concept. The problem is that scientists and others believe that infinite sets exist in nature. This is why we have crackpot nonsense like continuous structures, nonsense that even Einstein, Mr. Continuity par excellence, was beginning to doubt. An infinite number of points on a line and infinitely smooth surfaces are pure unmitigated hogwash in the not even wrong category.

The reason that continuity is still considered a viable concept in physics is that physicists are political (i.e., gutless) animals and nobody can say anything bad against mathematicians and retain their careers. Once you remove infinity from physics, all the infinity mathematicians become obsolete. The jackasses have retarded progress in science by centuries.

I’m still waiting for the resident materialist/Darwinist know-it-alls to gather enough huevos to answer the simple little challenge I posed. Einstein claimed that bodies moved in spacetime along their worldlines or geodesics. Answer the following question with a yes or a no. You can either agree with Einstein or disagree.

Can a body move in spacetime?

LOL. No takers?

Mapou #67

Well, I don’t know many objects or things in life that are infinite BUT some of the models we use are. For example, a sine wave extends out to infinity. The normal curves we use for probability distributions extend out to infinity (I know, the data doesn’t but the mathematical model, which is useful, does). Likewise the force of gravity doesn’t just end at some point, it extends out to infinity even if there is a practical limit. The solutions to differential equations that are used to model electrical systems extend out to infinity.

On the quantum scale entanglement doesn’t seem to be limited to some distance. And black holes are something like a singularity, i.e. what you get when you divide by zero.

I don’t think that’s true at all. In fact, in my experience, the relationship between mathematics and physics can be quite tense. And we’ve made the greatest breakthroughs in physics since Cantor did his work. I don’t think continuity has harmed physics at all.

Mapou #68

Don’t they alway move in spacetime?

Maybe I’m confused.

Jerad:

Nobody knows where the forces of gravity or electric fields end. Also, gravity breaks down at very small distances where the inverse square law predicts infinite gravity (at r = 0). If this were true, all particles would collapse on themselves. This is not observed.

This is obviously crackpottery of the worst kind since dividing by zero is nonsense.

That’s funny because I have not seen anything, let alone breakthroughs, derived from the assumption that nature uses either infinity or infinitesimals. Even the use of calculus, the so-called math of continuous structures which is presented as an example of infinity in practice, does not use infinity at all. In fact, I routinely solve calculus functions on my computer and I can assure you, my computer is as finite and discrete as can be.

You are not just confused; you have been deceived and taken to the cleaners. There can be no motion in spacetime whatsoever. This is why Karl Popper called spacetime, called a block universe in which nothing happens. However, I’ll delay my explanation of why there can be no motion in spacetime until somebody in the materialist/Darwinist camp can gather up the guts to offer a refutation of my position.

Zachriel,

“phoodoo: You do realize that numbers aren’t actually real things, right? What do you think the largest prime number is? Since numbers are just concepts, there is no largest number. Infinity is nether the largest nor the smallest number, since it doesn’t represent anything.

So close! You do realize that infinity is a concept, just like numbers are concepts?

alter-phoo: Infinity is nether the largest nor the smallest number, since it is not a number.”

Do you think there is some contradiction there? There most certainly isn’t. Lots of things that are concepts are not numbers. Big is a concept (its not a number!) The beginning is a concept. The end is a concept. Fruitless is a concept. Infinity is a concept, it is not a number. It can’t even be used in mathematics, because every answer it gives is nonsense. It is the same as saying 3 X Big=?

The equation is nonsense. Infinity is not a number.

Mapou #70

I don’t think the extent of the force of gravity or electric fields end at all. They just get weaker and weaker.

The inverse-square law for gravity always refers to the distance between the centres of the masses so r is never actually zero.

It’s analogous, not an exact comparison.

Well, as I said, the models used in physics extend out infinitely far. It’s true that there are lots of perfectly good finite numerical methods which can give you very, very good approximate results but

arcsin(1) = pi/2 +/- 2pi x n where n = 1, 2, 3, 4 . . . .

Not only are there an infinite number of exact solutions but part of the exact answers is pi which has an infinite, non-repeating decimal expansion. Any decimal expression used for calculations is just an approximation. When doing things like Fourier analysis we try and find exact answers which then can be approximated as needed.

Fourier analysis involves finding infinite sequences representations for sets of data. It’s used all the time in some areas of engineering.

Remember that many commonly used quantities (like pi and the square root of 2 and e) have infinitely long decimal expansions so you do use infinite things all the time!!

I’m not volunteering to refute your position, whatever it is. But surely all things that exist move through spacetime. My atoms are moving through time certainly and, owing to the motions of the earth, solar system, galaxy and universe through space. But, like I said, maybe I’m confused.

phoodoo #71

Except it is used in mathematics all the time. You can take the limit of a quantity to infinity and it’s quite common to take definite integrals out to infinity (which is defined to be a limit really). You add up infinitely large sequences all the time.

Mapou:The problem is not that infinity is not a viable mathematical concept.That’s not what you argued above.

Mapou:The problem is that scientists and others believe that infinite sets exist in nature.Don’t know too many who think that.

Mapou:In fact, I routinely solve calculus functions on my computer and I can assure you, my computer is as finite and discrete as can be.As pointed out above, computers can be programmed to work with the mathematics of infinity.

phoodoo:Infinity is nether the largest nor the smallest number, since it doesn’t represent anything.That is incorrect. In analysis, infinity represents an unbounded limit.

Jerad:

Great. We use the derived relationship to show one is a proper subset of the other. And we should use the same relationship for everything else. To use a contrived relationship is bogus, but you don’t seem to be able to grasp that simple fact.

If we take set Z+ and subtract set E from it, we get another infinite set- the set of all positive odd integers. That alone proves that set Z+ and set E do not have the same number of elements.

Jerad, good at following, not so good at thinking for himself.

Joe @ 74 – now label all of the positive odd integers (i.e. O=Z+\E) with their ranks. Aren’t their ranks just Z+? If not, which element(s) of Z+ aren’t a rank of O?

Bob, what is your point? I have already covered contrived relationships.

Joe #76

I searched for contrived mathematical relationships . . . not much pure math stuff came up. In fact, nothing really.

Yes, Jerad, I already know that you are very limited. Here try this:

contrived:: having an unnatural or false appearance or qualityIt means the same in math as it does to the rest of the world.

Joe #78

That’s perfectly good definition of contrived. BUT contrived has no specific mathematical definition. So if you say: that correspondence is contrived it only means that it seems a bit false. But it doesn’t mean it’s not valid!!

You have yet to show that contrived has been adapted for a specific mathematical meaning. And so . . .

Your complaints about contrived correspondences are trivial.

Jerad, The meaning of contrived remains the same regardless if it is used in mathematics or not.

More than a bit false and yes it is validly false.

Joe #80

Despite several requests you have failed to provide a mathematically specific definition of contrived. Which means you cannot MATHEMATICALLY object to this or that correspondence because there is no mathematical definition of contrived.

You painted yourself into the corner Joe. And then you failed to find the path out.

Whatever that means. In Joe world. And he’s not going to tell us.

I know no ID supporter on UD will criticise another supporter. Maybe it’s because you feel beset upon from all sides. But, seriously, if you can’t as a group decide what you do and don’t believe in then those on the outside don’t know whether to take someone like Joe seriously

Does Joe speak for you? Have a say!!

Jerad:

Why do you think that is a requirement? The definition of contrived is what it is.

Pure gibberish.

That’s your biased opinion.

Jerad @72:

Well you are. Nothing can move through time or spacetime by definition. This little inconvenient truth is understood by a few people in the physics community but you will not see it mentioned much anywhere because it makes a lot of the claims regarding Einstein’s spacetime physics look rather silly. I was waiting for some of my more strident detractors to take the bait but bravado is not synonymous with gonads in the materialist/Darwinist community. Here’s the simple reason that nothing can move in spacetime, which also why there can be no time dimension. The following is copied from a blog article I wrote more than four years ago titled Why Einstein’s Physics Is Crap 😀 :

Now I realize that most materialists/Darwinists prefer the opinion of authority than an actual simple proof. So I prepared the following:

Here are a few quotes from knowledgeable people regarding the impossibility of motion or change in Einstein’s spacetime:

There is neither space nor time. There is only the changing present. Time travel is for total morons. But don’t tell that to Stephen Hawking and his band of clueless followers because they’ll take offence. Enjoy.

OK UD- How many people think that when we look to see if one set is a proper subset of another that we use relationships that are derived. For example the infinite set {2,4,6,8…} is a proper subset of {1,2,3,4,5,6,…} because all of the elements in the first set are also contained in the second. Subsets are derivatives of the (super)set that contains it. There is a natural match of like numbers.

Does everyone agree with this? If not I am open to correction so please speak up

Joe #82

Why do you think that is a requirement? The definition of contrived is what it is.

You were using it in a specifically mathematical context, implying that it had a mathematical meaning. You contrasted it with ‘deriived’ but you HAVE NOT been able to provide a quote or context where that MATHEMATICAL distinction is made.

You have not shown how to MATHEMATICALLY distinguish between contrived and derived.

Time to stop dancing and to start proving your case.

Yes, I have. For example the infinite set {2,4,6,8…} is a proper subset of {1,2,3,4,5,6,…} because all of the elements in the first set are also contained in the second. Subsets are derivatives of the (super)set that contains it. There is a natural match of like numbers.

The derived relationship is what we use to determine whether or not one set is a proper subset of another.

All other relationships between the two sets are contrived.

And geez, I have been saying that for months. Time to stop with your willful ignorance.

Mapou #83

Whatever. I’m not sure I need to respond.

Joe #84

Clearly {2, 4, 6, 8, 10 . . . } is a subset of {1, 2, 3, 4, . . .} It’s the use of the terms derived and contrived that is in contention.

Joe has asserted (without citations) that derived and contrived are meaningful mathematical terms when working in Set Theory. I say they aren’t. I can’t find mathematical definitions of those terms in regards to Set Theory. I studied a bit of set theory and I don’t remember derived and contrived being important ways of distinguishing types of one-to-one correspondence.

Please be honest and judge this disagreement based on the mathematical literature.

Deal?

Jerad:

Only by people ignorant of the language.

LoL! Please be honest and judge this disagreement based on the commonly used definitions of the words and the context they are being used.

I say there is one derived relationship between two sets and we use it to determine whether or not on set is a proper subset of another. And all other relationships are contrived. I have made my case in comments 84 & 86.

That is and has always been my argument regardless of how Jerad wants to try to spin it.

Joe #86

Subsets are’derivatives’ of the set that contains it?

JOE, just provide a proper reference for your usage. You’ve been asked many times. Time to put up.

This has NO mathematical meaning.

Well, let’s deal with it once and for all. Provide your reference extolling your use of derived and contrived regarding Set Theory.

You’ve been asked, time to provide.

Jerad:

Of course not. Your opinion is not worth all that much anyway. 😀 See you around.

Jerad:

I reference the English language

Yes, a subset is derived from some other (super)set.

Do you understand the natural match of numbers between a subset and its superset?

Joe #88

But that gets to the exact point of the matter. Joe strongly implied that ‘derived’ correspondences’ were preferable to ‘contrived’ correspondences in a strictly mathematical sense. He has been unable to provide that strictly mathematical definition. I looked, I thought I might be wrong. I couldn’t find it.

When you have scientific/mathematical discussions then the terms you use can take on very specific/particular meanings and you have to be aware of that.

‘Wave’ means many different things depending on the context. And if you’re talking to a physicist they you’d best be precise. Most of you won’t know that ‘path’ is a rather specific term in an area of mathematics called Graph Theory. Derive does have a general definition and usage. But that doesn’t mean that when you use it in a mathematical context you can be so lenient. This happens in theology as well as some of you will know. In any discipline it’s necessary to get specific.

Joe is not using the terms ‘derive’ and contrived’ in any clearly defined, set theory specific way. And so they do not necessarily mean what he thinks they mean. And he is not providing any references for his interpretations.

And I say this is nonsensical. But Joe does a good job of standing his ground which I do applaud.

I continue to asset that the terms ‘derived’ and ‘contrived’ have no specific mathematical meaning in Set Theory one-to-one correspondences.

I say there is one derived relationship between two sets and we use it to determine whether or not on set is a proper subset of another. And all other relationships are contrived. I have made my case in comments 84 & 86.Yes, I know you are very limited. There is one natural match of numbers between a subset and its superset. That is the derived relationship. Period. And it doesn’t matter if Jerad gets all angry and stomps his feet.

Well I made my case and Jerad refuses to address it. I will leave it for people to judge for themselves.

Joe #91

But we’re talking mathematics. Not the same thing.

Why not say extracted or culled?

You used a word, you got called on it, and now you’re desperately trying to defend it.

I’ve never read/seen a definition of a ‘natural’ match of numbers between a subset and its superset. Please provide one.

And, please note, Joe has STILL not provided a mathimatical context or reference for his use of contrived and derived in a set theory context. Just saying.

Jerad:

They mean something in science. Biology uses those words. Why is mathematics immune to them? Does Jerad think that there isn’t any natural match of numbers between a subset and a superset? Really?

Jerad is stuck inside of his little box and his limited intellect won’t allow him to escape.

So is the match supernatural, nonnatural, pre natural or artificial? Or do you think your difficulty with the language means something?

I have provided a simple example. You choked.

Jerad:

I used a word, I defended my use and now you are desperately trying to hide your ignorance.

AGAIN:

Jerad:

Great. We use the derived relationship to show one is a proper subset of the other. And we should use the same relationship for everything else. To use a contrived relationship is bogus, but you don’t seem to be able to grasp that simple fact.

If we take set Z+ and subtract set E from it, we get another infinite set- the set of all positive odd integers. That alone proves that set Z+ and set E do not have the same number of elements.Jerad, good at following, not so good at thinking for himself.

Joe #93

And yet Joe STILL cannot provide any references to mathematical usages of his terms. Nor has he even tried.

But it does matter if you can defend your statements with proper academic references. You haven’t even tried.

Still, Joe has not established that his use of ‘contrived’ and ‘derived’ have any basis in Set Theory. And I’ve never heard of a general ‘natural match’ principle.

How can I address a ghost? Something with no corporeal grounding.

I implore all those who read this discussion to respond honestly. There is no point in having and sponsoring discussions if we’re not honest.

AND it would be nice if y’all didn’t just assume I’m a troll and I’m lying. It would be nice to be treated as a human being. Okay?

Joe multiple responses

I’ve asked you many times to provide your set theory specific meaning of those terms. You haven’t done so. What’s the problem?

No one is saying mathematics is immune but if the terms aren’t strictly defined in context then they have little explanatory power.

Nice to see that UD’s policy of open, honest and respectful discussion is being promulgated.

You can define and use any term(s) you wish Joe. You just have to be clear about it. You haven’t done that.

You have not been able to defend your use of the word in a specific set theory context. Please do so.

There are lots and lots of way of setting up a one-to-one correspondence between the positive integers and the postive even integers. It’s the fact that a one-to-one correspondence exists that’s pertinent. As long as we can line up the sets and match the elements one for one then we can decide about things like whether one set is a subset of the other or whether they have the same size.

Too bad that’s wrong.

If you’d rather be wrong that’s up to you Joe.

Jerad, The words mean what they do regardless of the setting. That you have some mental block to that fact says quite a bit about you.

I choose to defend my statements with evidence, logic and reasoning. Only you require the crutch of other people.

So what do you think the match is that is between the same numbers of different sets? Why are you so happy to live inside of your little box?

Well I made my case and Jerad refuses to address it. I will leave it for people to judge for themselves.Go get stuffed.

Jerad:

And only one is natural, meaning it directly matches the same numbers to each other. And yes, that matters.

If we take set Z+ and subtract set E from it, we get another infinite set- the set of all positive odd integers. That alone proves that set Z+ and set E do not have the same number of elements.Cuz Jerad sez so? Really??

I hate to harp on about anything but:

Joe has NOT provided any academic (or otherwise) references to how he wants to use the terms ‘contrived’ and derived’ in a set theory specific context.

I have tried my best to be respond to my objectors and, hopefully, at least as respectfully as they have responded to me.

It is very, very important that y’all respond to this discussion objectively. That is, don’t favour my or Joe’s arguments because of which ‘side’ we’re on. We were discussing mathematical terms. Respond based on what we said not who we are.

And please, check things out for yourself. Google search. Test things out. Don’t just take anyone’s word for it.

Make up you own mind. That’s all anyone can ask.

Jerad:

LoL! There are two different systems used. I am saying we have one and should use it for both. You disagree but cannot provide any reasoning.

Jerad:

I used examples along with the common definitions of the words. Jerad doesn’t like that and whines.

Hypocrite.

Set Subtraction:A way of modifying a set by removing the elements belonging to another set. Subtraction of sets is indicated by either of the symbols – or \. For example, A minus B can be written either A – B or A \ B.Hey looks like I was right about that! Jerad disagreed yet sez that I don’t know set theory.

Joe, multiple responses

In some specific contexts words can take on more specific and narrow meanings.

In English you can refer to the ‘gravity’ of the situation and mean something different from when you use that term in physics.

‘Charge’ in the military means something completely different from ‘charge’ in electricity.

When you use a term implying that is has a specific mathematical usage then you have to be specific about that usage. You have to provide the context. And you have not provided a solid, strict, mathematical definition of ‘contrived’ and ‘derived’ although you implied they had them.

This is not personal. And, if you’ve got the context, it should be easy to provide.

Welcome to science. Don’t block the doorways.

What if the sets don’t have the same numbers? Consider this:

Is there a one-to-one correspondence between the sets

{1, 3, 5, 7, 9 . . . } and { 2, 4, 6, 8. 10 . . . . `]

`

Joe gets technical.

And no, it doesn’t matter if all you want to do is show that the sets are the same size.

One thing at a time. You can’t even grasp the simple concepts I am discussing now.

That depends on whether or not you want to show they are the same size in a real or imaginary sense.

Then it’s strange that I just provided a reference that agrees with me. Weird, eh?

Jerad is on a jihad. Anything that does not come from the materialist Church is automatically wrong.

I, too, am on a jihad. Anything that comes from the materialist Church is suspect. And, as we all know, the materialist Church keeps piling on the crackpottery, nonstop.

Jerad:

I have provided the definitions for the words in the context they are being used. I have told you that the meanings of words don’t change just because we are using them to describe what is happening in Set Theory.

It’s late, I’m tired.

Despite my asking multiple times Joe will not or can not provide a suitable, set theory mathematical context for the terms ‘contrived’ and ‘derived’ as they influence evaluating one-to-one correspondences.

I am NOT on a jihad Mapou. I am merely asking that Joe be specific and discipline sensitive to an assertion he made. What’s wrong with that? This isn’t even a major topic. He just claimed that ‘derived’ correspondences were ‘better’ than ‘contrived’ correspondences and I’m trying to figure out what, specifically he means.

Joe, when you get around to actually addressing this terribly simple request I’ll respond. And if you’re not going to bother then don’t waste my or your or anyone else’s time. Time to face the music and dance or leave the floor. It’s up to you.

(I’m pretty sure I know what’s going to happen; Joe is going to buster and bluff a bit more and hope everyone forgets that he dodged a basic question about an assertion that he made. If that’s okay with everyone else then I’ll stop pursuing it because I’ll know that this forum is NOT open and fair and transparent. I’ll know that there’s a double standard. But, I’d really love to be proved wrong.)

Jerad says,

I’ll stop pursuing it because I’ll know that this forum is NOT open and fair and transparent. I’ll know that there’s a double standard.

I say

What???

If Joe does not agree with you it proves this forum is not fair and transparent??

I would think that sort of thing would be evidence of an open fair and transparent forum?

For the record

It looks to me that Joe and Mapou are all wet on this one.

I have no idea what point they are trying to make. I think that it has something to do with infinity not existing in the phyiscal universe.

I would agree with this as far as it goes and I don’t know anyone who would disagree.

Not existing physically does not equal not real however.

Mathematics does not have to have a one to one correspondence to the universe to be true and valid.

peace

nice summation fifth

fifthmonarchyman,

Infinitesimals are assumed in Einstein’s physics. It is called continuity. This is the reason for the term ‘spacetime continuum’. General relativity, for example, is based on continuous structures, i.e, infinitely smooth structures. This is a fact. So much so that Einstein had doubts about its correctness as I pointed out earlier:

Mapou,

I agree that infinities do not exist in the universe and I agree that Einstein’s physics is at best very incomplete.

However You and Joe seem to be arguing something much stronger than that. You seem to be saying that infinities are not real. I can’t agree with that.

Physicality and reality are not synonymous

Does that make sense to you?

peace

Jerad:

And I say that I have done so.

Look I have explained my position such that middle school students understand it. The way I see it the problem is all with Jerad.

Again the example:

Here’s an infinite set, let’s call it Z+ = {1, 2, 3, 4, 5 . . . .}Here’s another infinite set E = {2, 4, 6, 8, 10 . . . . }Set E is a proper subset of set Z+ due to a natural match of same numbers. That is it is a derived relationship of a one-to-one correspondence.

To take those same two sets and say that the element placement/ rank can also be a one-to-one relationship (a mapping function is used) is then a contrived relationship.

FMM:

Right now I am saying that the set of all positive integers has more elements than the set of all positive even integers.

So the military never uses electricity? You can be charged with a crime in the military. You can charge things on a credit card in the military also.

Mapou,

I’m sure you are also aware that Newtonian mechanics does the same thing, that is, it models space and time via smooth manifolds. Is that an issue for you?

Joe #118

And that is wrong. It’s been known to be wrong for over 100 years. And yet no one but me points this out.

If you guys want to be stuck with 19th century mathematics it’s okay with me. But I’m disappointed.

This is not controversial. This is not something that people argue about (anymore). There is nothing about this that is part of the ID v evolution debate. There is no reason NOT to call Joe on this.

But no one does.

By the way this:

is gobbly-gook.

If this is an example of your collective ability to understand mathematics then you’ve already missed the train. How do you expect to even begin to grasp some of Dr Dembski’s points if you don’t get this?

Here’s a question. How can one infinite set be bigger than another infinite set if both have no end? That’s the problem with infinite sets. They only exist in the imagination of mathematicians and materialists.

Fifth:

No, it does not. I’m a yin-yang dualist. I believe that reality consists of two complementary opposite realms, the physical and the spiritual. The former can be created, destroyed or modified. The latter can neither be created, destroyed or modified; it just is. IMO, neither the physical nor the spiritual contains infinite sets of anything. Infinity is an abstract concept. Abstractions do not exist.

Mapu #121

That’s what Cantor established in the late 19th century. His work was extremely controversial at the time but eventually it was shown to be sound and is now part of the structure of mathematics.

Let me try the ‘ID way’ : The creator created Earth as a sphere (if you ignore the ‘bulge’ due to spinning ,which forms a oblate spheroid). Sphere is possible because of a point on infinite plane, so you see, infinity is what allowed creator to create the Earth.

Jerad:

I don’t care what Cantor established. It is still hogwash.

Mapou #125

Seriously, don’t expect anyone with a modicum of a mathematical background to take you seriously again. This is well established, non-controversial stuff. I learned it at the undergraduate level.

It’s your call but I imagine the folks at The Skeptical Zone will find your position amusing.

Jerad @ 126

When Mapou doesn’t even care what Cantor computed, why would he care what folks at TSZ think ?

BTW, Cantor’s Diagonal argument applies to

anysetMe_Think #127

It’s his call.

Yup but pointing that out here is a waste of time ’cause, apparently, Cantor was wrong.

Mapou says

The latter {spiritual}can neither be created, destroyed or modified; it just is. IMO,

I say

You are entitled to your opinion just not your own facts 😉 You sound like a Mormon and not a yin-yang dualist.

You say,

Abstractions do not exist.

I say,

from here

https://www.google.com/search?q=abstract&ie=utf-8&oe=utf-8

quote:

Abstract : existing in thought or as an idea but not having a physical or concrete existence.

end quote:

Are you saying that something has to be phyiscal or concrete to exist? You do realize that is the position of the materialist. Is love an abstraction? What about morality?

Joe says

Right now I am saying that the set of all positive integers has more elements than the set of all positive even integers.

I say,

check it out

http://www.businessinsider.com.....ty-2013-11

peace

F/N: A few balancing words, as we count down the hours to the pre-Christmas Assembly sitting here.

(Forgive the personal note, which is why I have been busy elsewhere. And no, he who abused privilege to slander me will most likely not be present; so the correction on record will be postponed. Cf. here, if you need to know more.)

Now, similar to how x^0 = 1, it is reasonable to accept zero as an even number. Indeed, it has properties that fit with that, and it is in the right place in the number-line to be even: even-odd-even-odd, etc.

(This is probably the easiest way to make it palatable to young students. Besides, it begs us to extend to negative numbers and complex ones. Which reveals my not so hidden agendas here.)

Likewise, because it is useful, we can define transfinite numbers of countable and beyond countable cardinality. Aleph-null is countable, and we can profitably reason:

We can go to power sets, and to the issue of the continuum, which is a transfinite number of higher cardinality than the countable one.

These are of course concepts, we have here a basis for defining that the continuum is such that on a number line, for ease of reference, between any two neighbouring values, we may define a third. And, we can note that abstract entities such as 2, 3, 5 and the relationship 2 + 3 = 5 necessarily exist in any world. Based on constructing {} –> 0, {0} –> 1, {0, 1} –> 2 etc, and defining addition and equality appropriately.

From this, we may extend to real world observations as a useful model. We do not need to make any absolute commitment to there being an actual continuum in the physical world, it is fine grained enough to use continuum models in many cases.

Similarly, by defining hyper-real numbers and taking reciprocals, we can give mathematical substance to infinitesimals and justify treating dx, dy, dz, dt etc as tiny all but zero numbers. Non-standard analysis, an alternative foundation to Calculus, lies down that road. (More of my not so hidden agenda . . . and I see Calculus is now in 5th form math in the UK, hint hint hint. For crying out loud, I have long known that it was reported decades back that every High School child in Russia has to do several years exposure to physics and calculus. Can’t we do an introductory Physical Science, Physics and Chemistry sequence and can’t we put some serious math in too? And some logic calculus? Tossing in some computer science too, Hon Mr Santa Claus, sir?)

Of course the usual ZFC set theory anchored, epsilon and neighbourhood, limits oriented approach to calculus can be taught as developing another enriching perspective. (My favourite demonstration is water flowing into a bucket. Add in a leak and we are off to system modelling and controls, thence of course differential equations, Laplace transforms and the complex frequency domain; also the discrete state Z transform equivalent and my favourite heavy rubber sheet with poles and nailed down zeros graphical model. Where, drop off the transient terms and we are in Fourier transform space. Another not so hidden agenda.)

Why am I doing this?

First, to remove the strawman caricature being painted above by those who pretend and prefer that supporters of design theory are scientific and mathematical ignoramuses as a whole.

And, BTW, when I take a bit of time, I want to revisit the Abu 6500 c3 reel as an example of FSCO/I and how it can be informationally quantified using the nodes arcs wiring diagram framework. As in:

In short, we here have a descriptive language reducible to bits through a chain of Y/N structured questions. As Orgel stated in 1973, without referring explicitly to bits.

That is, the shoe is on the other foot.

Then, we can look at the ribosome etc as cases in point of FSCO/I.

And so forth.

KF

PS: Remember, how, over the course of several weeks, too many objectors to design theory, were unable to bring themselves to acknowledge that FSCO/I is real, that it was conceptually identified by Wicken and Orgel in the ’70’s as applicable to life, and that it points to a way to quantify the information content of organised functional entities, at least in principle based on structured chains of Y/N questions. Never mind what AutoCAD etc routinely do. As for that such FSCO/I extends to cases in the world of life . . .

KF #130

Thos are infinite by the way Joe.

Your flock needs some learnin’ regarding Cantor’s work.

Jerad,

With all due respect, you just used loaded, inapplicable terminology — “flock” — that insinuates mindless following; when in fact we see understandable skeptical concern because of widespread and longstanding major violations of trust in institutions that should be paragons of intellectual virtue.

As in, there is a reason why the term junk science has arisen, and why people reasonably fear Government backed ideologised institutional science.

Let’s start with: E-U-G-E-N-I-C-S.

As in, that’s no phobia.

Or, have you forgotten that Hunter’s Civic Biology, at the heart of the Scopes trial, was riddled with eugenics?

One wonders how comes that inconvenient little fact tends to at best be lost in the footnotes . . .

That, eugenics laws were put on the books, with horrible consequences, and not at all just in Germany?

Indeed, Germany explicitly copied US laws, IIRC esp. those of California?

Do you see now, why for cause people may be inclined to be fairly suspicious?

When trust is abused, it will be forfeited.

As Climate Scientists are currently finding out.

And, as is building up on origins science.

And when trust has been sufficiently abused on a large enough scale, it poisons the atmosphere, leading to questioning anything that seems counter to common sense.

That can become an excess, but the need tyo be critically aware in an age of widespread ideological abuse of science should be frankly faced.

Instead of sneering contempt and loaded insinuations, it would be more reasonable to ground and warrant.

But then, isn’t that what I have been calling for on origins science for two and more years in reply to some remarks you and others made?

With no serious takers to date?

Now, I highlighted that Mathematical models, including continuum hold primarily, conceptual validity. They are extended to real world contexts by use of models and meaning assignments. That this works very well is an astonishment to those whose fundamental view of nature is non rational and arguably irrational.

But, from the days of Newton, Boyle and Kepler, that is no surprise to those who understand our cosmos to have been shaped by an utterly rational architect.

I suggest, lastly, that when one refers to actual numbers, transfinite is a more applicable term, with understanding that such numbers start scaling with aleph-null, the cardinality of the natural numbers and linked sets.

KF

Hey KF

You put your finger on a phenomena that is pervasive in our world today. The complete collapse of the old authority structure

Secularists thought that when people abandoned religious authority it would mean that their own authority figures (scientists and government officials) would replace it.

What is happening instead is anarchy. Each person trusting only those who agree with him.

You see it on both sides of the isle. The materialists here won’t even look at arguments from anyone who is not a fellow traveler for example.

Talk about Déjà vu

quote:

In those days there was no king in Israel. Everyone did what was right in his own eyes.

(Jdg 17:6)

end quote:

We all know what comes next

peace

KF #133

I thought you’d like the Biblical allusion. Sorry.

Kind of off topic don’t you think?

About stuff they can easily look up for themselves? That’s part of most undergraduate mathematics curriculum?

If they didn’t trust me they could just do a search for themselves or look it up in Wikipedia.

Waaaaaay off topic now.

Definitely a minority opinion.

You mean like an underfined, undetected designer?

I’m sorry, that was a bit snide. But you get my point? You call me out when I question your beliefs but you defend you compatriots when they question me over something that isn’t even controversial.

I know, you’re going to accuse me of inflaming the situation. But I’ve spent hours trying to explain this stuff which you clearly understand only to be told its hogwash. That such things exist only in the minds of mathematicians and materialists. What?

How about this from Mapou #110

Very understanding and respectful eh?

I spent hours being polite, trying to explain, here and on Joe’s blog. And, in the end, I’m told its wrong.

LOOK IT UP!!

Yes . . .

A matter of opinion. I think relativity and quantum mechanics shook things up a bit.

From Wikipedia:

“Transfinite numbers are numbers that are “infinite” in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as “infinite”. However, the term “transfinite” also remains in use.”

So, it was just a term used to ease the ‘pain’ of accepting his work. I don’t know what you mean by they ‘start scaling’. Scaling means making bigger or smaller but it’s pretty clear that now most people accept that transfinite numbers are really infinite.

Jerad, Session now in progress, complete with a lead in by the speaker that tried to justify himself but implicitly conceded my point, looks like the intervention of the Christian Council about angry remarks and attacks was decisive. No time for a full response. I pause to note that transfinite avoids the issue of an actual infinity, which is still a problematic matter, especially for those who imagine they can deliver an infinity in countable succession. As in an infinite, causally successive actual past of inherently contingent beings. KF

PS: I note that the eugenics issue, the Scopes Monkey trial, Hunter’s Civic Biology and linked issues are highly material to the polarised debates that surround origins science, and lurk behind the subtext of contempt that so often crops up when loaded allusions used by objectors. So, if there is an attempt to project the pall of obscurantist fundamentalism, such are immediately, highly relevant. And, I will not shy away from pointing out that fact.

PPS: Try the sequence of patently increased cardinality aleph null, power set thereof, and so forth, with the debate as to what c the continuum cardinality is, often suggested — note the term — to be that of the power set of aleph null.

jerad,

Oh please, to suggest that all mathematicians agree that some infinities are bigger than others (much less other thinkers besides so called mathematicians), is pure nonsense. Plenty of mathematicians don’t think infinity is a real concept.

If infinity means something going on forever, then playing with words to call one unlimited amount bigger than another unlimited amount, is just playing with words. Its no different than saying “the biggest thing possible”, and then claiming the biggest thing possible plus a little more is even bigger. Cantor doesn’t get to make the illogical logical. Its pure word play bullshit.

If I say 2 x infinity is bigger than 1 x infinity, that doesn’t make it so.

phoodoo:If infinity means something going on forever, then playing with words to call one unlimited amount bigger than another unlimited amount, is just playing with words.It’s a matter of mathematical proof, not opinion. See Cantor 1874 or Cantor 1891.

Not its not Zachriel,

there is no mathematical proof, if the definitions of your symbols are rubbish.

Is 2 x infinity bigger than 3 x infinity? Can I claim a mathematical proof says so?

Mapou,

Here’s one way to look at it: Some sets, such as the integers or the rational numbers, can be arranged in an (infinite) list. This diagram shows one scheme for putting the positive rationals in a one-dimensional list.

For other sets, such as the real numbers, this is impossible. Any list of real numbers which you construct will be missing some values. In that sense, the set of real numbers is larger than the set of rationals or integers.

My post #120 was in moderation for a little while; I’d be interested to hear your response.

phoodoo:there is no mathematical proof, if the definitions of your symbols are rubbish.Well, take a look at Cantor 1891 (the diagonal proof) and tell us where he went wrong. Generations of mathematicians have found the proof convincing.

phoodoo:Is 2 x infinity bigger than 3 x infinity?No.

DaveS (Zacheriel, this is where Cantor went wrong),

What does “missing some values” in your post mean? Missing from what?

That diagram is totally artificial. There are no requirements for what must match up with something from another set.

One set is a bunch of things you can label anyway you want, and another set is also a bunch of things you can label a different way if you choose. There is no universal requirement that one label matches another, and that therefore you can say some labels don’t have a counterpart in the other set.

phoodoo:There is no universal requirement that one label matches another, and that therefore you can say some labels don’t have a counterpart in the other set.In set theory, a bijection, or one-to-one correspondence, is how we determine if two sets have the same number of elements. We can prove that rationals and the natural numbers have a one-to-one correspondence. We can prove that real numbers and the natural numbers do not have a one-to-one correspondence. In other words, there are more real numbers than can be counted with natural numbers.

phoodoo,

By “missing some values” I mean that given a list of real numbers, you can construct another real number which is

noton the list. Therefore the list was incomplete.I don’t know what you mean by “artificial” there. The diagram simply shows one way to arrange all (positive) rational numbers in a one-dimensional list. There are infinitely other ways to arrange them, all of which work just as well.

Try as much as you like, you will never find such a list which includes all the real numbers.

Zacheriel,

The bijection is totally artificial. I could just as easy color all of the elements in one set red and all of the elements in the other set blue. Then I just match one red with one blue for eternity. Problem solved

Since infinity is a made up concept, which has no tangible reality, it makes just as much sense to ask, If I have peace in one set, and beauty in another, which is bigger?

DaveS,

The reason that the whole argument is an artificially constructed one is because who do you think named the different kinds of numbers? Who decided that some numbers are going to be called whole, and some natural, or real or complex or integers….? Those are completely made up names, you can just as easily call any number you can think of Fred.

So now you have several sets, one is an infinite number of Freds, another set is an infinite number of Freds, plus an infinite number of M&M’s. A third set has an infinite number of jelly beans, plus an infinite number of M&M’s but no Freds. Which set is biggest?

Answer: None

phoodoo:I could just as easy color all of the elements in one set red and all of the elements in the other set blue. Then I just match one red with one blue for eternity.However, unless you specify the order you place them, you don’t know if you have matched them all up, so you haven’t shown whether or not there is a one-to-one correspondence. However, if we arrange the rationals diagonally, we can show that we have included all the rationals, and that they can all be matched to the natural numbers.

phoodoo:Those are completely made up names, you can just as easily call any number you can think of Fred.Sure, though we might be more specific and say any number phoodoo can think of is a phoodoo number.

However, we’re not talking about phoodoo numbers, but natural, rational, and real numbers. Each of these types of numbers have unambiguous and explicit mathematical definitions, but you probably intuitively know at least the natural numbers; 1, 2, 3, …

Who do you match with whom if I call numbers Fred?

But Zachriel, the definitions are man made, for crying out loud.

Which set is bigger, the set of infinite M&M’s or the set of infinite M&M’s plus infinite jelly beans? Or the set of infinite candy? Jelly beans and M&M’s and candy all have definitions.

phoodoo:Who do you match with whom if I call numbers Fred?Don’t know Fred. Can he count? Can he do fractions? Decimals?

As for your red and blue matching:

Z: unless you specify the order you place them, you don’t know if you have matched them all up, so you haven’t shown whether or not there is a one-to-one correspondence. However, if we arrange the rationals diagonally, we can show that we have included all the rationals, and that they can all be matched to the natural numbers.

Right now I am saying that the set of all positive integers has more elements than the set of all positive even integers.

Prove it.

Perhaps to your limited intellect it is. However your limited intellect means nothing to me.

phoodoo,

Well, you can call it artificial if you like, but over the millenia people have found it useful to distinguish between different types of numbers. For example, numbers which can be written as a fraction of integers have nice properties that arbitrary real numbers do not have in general. Integers are even nicer, IMHO.

Obviously not enough information to go on here! Fortunately a lot (but not everything) is known about the real numbers and certain of its subsets, so comparing the ‘sizes’ of the reals and the rationals is possible.

I’ll be offline for some time, so have fun!

Funny how IDers are hell bent on proving God can’t be omnipotent since there is no infinity (and hence there can’t be a creator with

infinitepower) 🙂Zachriel

Fred = All numbers

M&Ms= M&M’s

Fred = Infinity

Infinite M&M’s = Infinity

Infinite M&M’s + Fred = Infinity

Infinity = Infinity

There is your mathematical proof.

It doesn’t matter what you call all of the population of an infinite set.

Cantor didn’t establish anything and no one has shown what he sad to be sound. t isn’t even of any use.

Joe @ 157 as of 10:24 am

Joe is ‘infinitely’ tipsy.

me thinks,

The only reason this discussion has any value to me, is simply to show how many so called skeptical thinkers are just willing to accept any old bit of nonsense someone claims, as long as they think that someone is a “scientist” or oooooo, a “mathematician!”

If Cantor says two infinite sets are different, well then, little be it for you to question, it just must be true. You never even bother to think who gave numbers that name in the first place. You can substitute any name for something else.

An infinite number of M& M’s is no different than an infinite number of blue M& M’s or an infinite number of Red M&M’s plus blue M&M’s. Just because I start adding more names to one set than to another, that doesn’t make one set bigger. Clearly if I used Cantors logic, an infinite number of blue and red M&M’s would be more than just an infinite number of blue M&M’s because there is nothing to pair up the red M&M’s with.

It just goes to show you how little people actually think, but instead just accept the word of authority.

phoodoo:Fred = All numbers“All numbers” is not well-defined. Do you mean the real numbers or the natural numbers or something else?

phoodoo:Infinite M&M’s = InfinityNo, that is not correct, otherwise an infinity of shoes would equal an infinity of M&Ms, which is clearly not the case. Rather, they have the same cardinality. That would be written | infinity of M&Ms | = infinity or | infinity of shoes | = | infinity of M&Ms |.

phoodoo:If Cantor says two infinite sets are different, well then, little be it for you to question, it just must be true.No, it’s because Cantor *proved* that the real numbers do not have a one-to-one correspondence with the natural numbers.

Joe @ 153 –

I gave one approach in 75, by using ranking. Another way to do it is to divide the positive even integers by 2. You then get back to the positive integers.

Infinity is tricky, and I certainly don’t understand a lot of the subtleties, but I wouldn’t reject areas of mathematics just because I don’t understand them.

Me Think – Obviously you are confused as I never said anything about infinity.

Bob O’H:

And I refuted that approach.

I refuted that also.

It’s as if you are proud to be willfully ignorant and you think it means something to the debate. Strange.

@Bob O’H

What set theory are you talking about? Joe has extended the set theory with his own definitions. So if you’re not talking about “Joe’s set theory”, then your approach is irrelevant.

Zacheriel,

Which set is bigger, an infinite number of blue M &M’s or an infinite set of blue and red M&M’s?

Cantor proved nothing.

JWTruthInLOve:

Please do tell as I am unaware that I am doing such a thing. So I would say that you are making that up.

phoodoo:Which set is bigger, an infinite number of blue M &M’s or an infinite set of blue and red M&M’s?They are the same cardinality as there is a one-to-one correspondence between the sets.

phoodoo:Cantor proved nothing.You say that, but haven’t shown it. Generations of mathematicians have found Cantor’s proofs convincing. It will take more than handwaving to discount it.

From my understanding infinity isn’t a number, it is a journey. Now if you are on a train (Einstein’s train) that had two counters, counter A counted every second of the journey and counter B which counted every other second of the journey. Counter A would represent the set of positive integers and Counter B would represent the set of all positive even integers.

At every second throughout the journey Counter A would have a higher count than Counter B, meaning set A would always have more elements than set B.

What my detractors seem to be saying is that counter B should count double every time it counts to allow it to be the same as counter A and because of that they are equal at every other second, and that is the only time you are allowed to look.

Zachriel:

His “proofs” are contrived and not derived. For example the derived relationship is that which is used to determine if one set is a proper subset of another. The contrived relationships disregard the natural matchups and because of that show a false relationship.

Zacheriel,

If you draw a straight line with all the blue M&Ms on one line, and another line with all the blue M&M’s from the second set and a third line with all the red M&M’s, you can match all the blues from one set with all the blues from the other set, but you still have all the reds left over, so clearly the red and blue set is bigger, right?

Its the same silly logic.

@Joe:

phoodoo:If you draw a straight line with all the blue M&Ms on one line, and another line with all the blue M&M’s from the second set and a third line with all the red M&M’s, you can match all the blues from one set with all the blues from the other set, but you still have all the reds left over, so clearly the red and blue set is bigger, right?There is a one-to-one correspondence. For each M&M in the first set, you map alternating between the second and third sets. To show the sets are different sizes, you have to show there is no such one-to-one correspondence, not merely that you can’t think of one.

Joe,

Could you provide a link or name the source from which you learned the concepts of contrived and derived relationships in set theory? You have piqued my curiosity. Thanks.

DaveS:

Certainly. Newton was a big fan of the continuity nonsense but he did not start it. The Greeks started it thousands of years ago with their silly concept of a line having an infinite number of points. Newton was a genius but he did not know everything. I don’t think he even realized that his inverse square law broke down at short distances. But he was right about the universe being absolute. He also understood that there was a force or causal principle responsible for inertial motion. He could not prove it, so he attributed it to God and ignored it in Principia. In this respect, Newton (and even Aristotle before him) was light years ahead of the relativists and everybody else.

Zacheriel,

There is only two sets. One which contains only blue, and one which contains blue and red.

There is a one to one correspondence of blue M&Ms to blue M&M’s.

The reds are all extra. Clearly a bigger set! Magic!

Jerad:

LOL. I don’t give a rat’s asteroid. And I am nobody’s female dog. 😀

The idea that two infinite sets can be compared to see which one is bigger than the other is stupid on the face of it. The only way to compare two series is to know their sizes. An infinite sets has no size that anybody can put a finger on. These so called one-to-one comparisons of infinite sets is ludicrous. Only materialist morons will swear up and down that they have anything to do with sizes.

Folks, got a moment to follow up. Let’s start with a basic point, Mathematics is an intellectual, logic puzzle game that did arise from practical considerations but has gone well beyond that. Probing and arguing has led in many fruitful directions, some of which were not conceived of when the original ideas were put down. And, it turns out that mathematical models have great utility in dealing with the real world. For example infinitesimals and calculus, whether we go by standard or non standard approaches. In short, lighten up. By certain accepted rules, it makes sense to speak in terms of the continuum being of higher cardinality than the natural numbers, it being not possible to put them into one to one correspondence. Yes, strange oddities do come out, but they are accepted on the implied logic. And let us remember, at no one point can we show our axioms to be mutually coherent, much less can we set up sets of axioms that are coherent that also point to all true claims in the field, by whatever notion of truth is relevant at this point, perhaps that on other axioms, they would follow. Bottomline, lighten up on Math. And, while many of us are forced to accept Math claims on authority, there is a context of a discipline that is a lot more rigorous than say, origins narratives. KF

PS: That God is not limited other than by the sort of things that are incoherent [God cannot make a square circle out of a paper clip . . . ], or inconsistent with his maximally great being character, has nothing to do inherently with the issues of transfinite numbers and infinitesimals, what continuity is or is not, etc.

I didn’t want to give the impression I was abandoning the conversation but for the next few days I only have an iPad and, lovely as they are, without a keyboard I find copying and pasting large quantities of text painful in the extreme. I’ve pretty much said my piece.

I’m sure the arithmetic of ‘infinities’ sounds very bizarre at first but it IS well developed and non-controversial now. And it is not just a few pointed-headed geeks with no tans or lives saying it. I find it much easier to grasp than quantum mechanics or relativity

Any set which can be lined up with the set of the positive integers so that each element of each set is uniquely matched up with an element of the other set (a one-to-one correspondence) is said to be countably infinite. So the positive even integers is the same size as the positive integers as is the set of all the primes and the set of the rational numbers and the multiples of three, etc. And, yes, if you take the positive integers and take away the evens you still have a countably infinite set of odds left. If you take the positive integers and take away the multiples of 4 you still have a countably infinite set. This may feel contrived or mere assertion but it’s not. AND you can look it all up online!! It’s not hidden or part of an agenda. Dr Dembski will be very familiar with this as will Dr Sewell.

Anyway I won’t be around much over the next few days. I hope you all have a nice Christmas!! I apologise for any typos or garbled text.

Joe @ 162 – I didn’t see any refutations, just something about things being contrived not derived. But I can’t see what the mathematical definition of “contrived” is, so I don’t know what your proof is.

I’m curious – are you saying that if I take every positive even number, and divide it by 2 I don’t get the integers? Can you tell me which positive even number this isn’t true for?

Hey all,

I’m amazed at the disagreements we are seeing in the area of mathematics.

this sort of thing should not happen

check it out

http://xkcd.com/263/

😉

peace

Readers with an interest in ‘Joe Math’ may enjoy this:

A lesson in cardinality for Joe GMy favorite bit of Joe Math:

phoodoo:There is only two sets. One which contains only blue, and one which contains blue and red.Yes, we understand, but you divided the second set into two sets, each of which are presumably infinite.

phoodoo:There is a one to one correspondence of blue M&Ms to blue M&M’s.Your assignment left out elements, but you might be able to append them to the end after the ellipses. There’s no way to know based on your assignment. We showed you how to form a one-to-one correspondence so that there is no ambiguity.

Mapou:The idea that two infinite sets can be compared to see which one is bigger than the other is stupid on the face of it.Cantor showed that the set of real numbers can’t be mapped in a one-to-one correspondence with the natural numbers. They are a bigger set.

Mapou:These so called one-to-one comparisons of infinite sets is ludicrous.That’s how we compare the size of sets, is through a one-to-one correspondence.

Zacky:

Nonsense. Correspondence has nothing to do with the size of an infinite set since there is no size. You people can delude yourself till kingdom come but the truth is the truth.

I don’t care how you compare the size of sets. It’s BS on the face of it. In my computer programs, I simply compare their lengths. This is the way it’s done by logically minded people. Wake up, Zacky-O.

Mapou:Nonsense. Correspondence has nothing to do with the size of an infinite set since there is no size.Handwaving is not convincing.

Mapou:In my computer programs, I simply compare their lengths.That doesn’t work with infinite sets, of course.

Zacky-O:

LOL. Convincing to whom? To bozos? You do realize I don’t give a rat’s asteroid, don’t you?

Of course it doesn’t and that is the point.

Mapou:Of course it doesn’t and that is the point.So because your finite machine can’t count to infinity, infinity doesn’t exist? Then why did you use the term “ad infinitum” above?

Zacky-O:

“ad infinitum” is a term that means that something can grow or extend indefinitely. It does mean that infinity exists. All sets are finite, period. Wake up Zacky-O.

PS. Calling a set infinite does not make it so.

Mapou:“ad infinitum” is a term that means that something can grow or extend indefinitely.It means literally to infinity, infinite meaning not finite, or without limit.

Mapou:All sets are finite, period.So the natural numbers are finite?

Zacky-O @188,

Natural numbers can grow indefinitely but that does not mean that you can have a set of natural numbers. If you can’t create a set on any conceivable computer, it does not exist.

Mapou:If you can’t create a set on any conceivable computer, it does not exist.So now it whether the computer is conceivable. We can conceive of an infinite computer, so that’s not a problem.

ETA: For instance, Turing conceived of “..an unlimited memory capacity obtained in the form of an infinite tape marked out into squares, on each of which a symbol could be printed.”

Zacky-O, the idea that either you or Turing can conceive of an infinite computer is 100% bogus. Your brain is very finite and, in your case, very limited.

Try conceiving of an infinite number of points on a line. Count every single one of them and report back to me when you’re done. I’ll wait.

The following set is finite, thanks be to the first cause:

{Joe, Mapou, phoodoo}

Mapou,

I’m on a tablet, so my apologies for not cutting and pasting. Regarding two of your posts:

1) What do you propose be done about the use of continuous structures in physics? Do you have an alternative to slot in its place? I think most physicists and engineers understand that these are just models; a straight train track is not literally an interval in the real numbers, for example. And people are able to land spacecraft on comets, after all.

2) Regarding the infinite set issue, you can represent both the integers and the rational numbers on a computer as lazy lists. You cannot represent the real numbers in this way, however, so that’s one way to distinguish between countable and uncountable sets in a computer context.

Zacheriel,

I just proved you wrong, and it was simple as could be. There is no ambiguity between matching a blue with blue.

Since there is nothing to match the reds up with, the set with reds must contain more right? Its simple, its obvious, and its completely stupid to suggest that just because one can arbitrarily decide to name something on one set, and thus only the ones with the names you prefer can be matched with the other set, that this means one set is bigger.

The whole concept of what you pair with what is an artificial construct, that anyone can do with any set, so its meaningless. But it does have meaning in showing that scientific materialists are as gullible as they come, just because they believe someone with a title tells them something.

phoodoo @ 194

If you understand the concept of line and line segment, you should understand infinity. Line has infinite length, line segment is part of the infinite line.

daveS:

Actually, nobody in physics uses continuous structures even if they think they do. Landing on comets is not proof of infinity. Calculus is as discrete as can be, a million mathematicians and physicists jumping up and down and screaming otherwise notwithstanding. Anything that uses numbers is discrete because numbers are discrete by definition. The problem is in the refusal to admit that the universe is necessarily discrete. It’s a disastrous mindset because it prevents us from seeing nature as it is. Once you accept that the universe is discrete, you immediately realize that distance (space) is abstract, a perceptual illusion: it does not exist. And it’s not just space. The concept of a time dimension in which we are moving in one direction or another is also a conceptual disaster. It is what prevents us from understanding why nature is probabilistic. I mean, if there is no time, it is impossible for nature to calculate the exact timing of interactions. Nature is forced to use probability in order to obey conservation laws in the long run. This is the reason that particle decay is probabilistic. It has nothing to do with such silly notions as state superposition.

Not true. Nobody, I repeat, NOBODY, can model the infinite. It is a form of nerdish delusion, pure unmitigated hogwash. Why? Because whatever model you use is finite.

I understand the concept of a line just fine me_thinks.

Now do you understand, the concept of a human creating what matches in one set with what matches in another set? It is a convenience. It is a way to organize ones own understanding. But it has nothing to do with the actual elements in that set.

One doesn’t have to use Cantors preferred way of choosing what matches with what. One can match the number 5 in one set, with the symbol.333333333 if that is what one chooses. Next they match -6.2 with 1/2.

There is no reason to prefer one matching system with another, other than it might make it easier for you to remember. As long as both sets contain an infinite number, you can always match one with another for infinity.

The sets contain an equal amount.

Some concepts in math are simply created to make it easier to explain a concept, that doesn’t make them reality or fact.

Is a point in geometry infinitely small? If it is, it is also infinitely large, therefore it has no size. You can not measure something with no size, so we must create size in our brains, to help to understand reality. Convenience and reality are two different things. Don’t let others always do the thinking for you.

the sizes of keiths in my brains are zero

Calculusis discrete? You have a very broad definition of discrete. But that makes me wonder what your issue with physicists using smooth manifolds is, if they are discrete objects. Sounds like the perfect structure for modeling a discrete universe!daveS:

Physicists do not use smooth manifolds even if they think they do. They use a discrete and finite computer to do their calculations. It would take an eternity to calculate smooth manifolds for the simple reason that infinite smoothness doesn’t exist and cannot be computed. As soon as one gets close to the discrete Planck scale, all calculations that assume smooth manifolds begin to break down. Take the inverse square law of gravity, for example. It breaks down when r approaches 0. If it didn’t, all particles would collapse under the force of their own gravity.

phoodoo:

That’s almost Joe-worthy.

Let’s suppose an electrical circuit is associated with the equation

[3sin(10 – t)]/(10 – t)

And you want to know what’s going on at t = 10. If you plug in t = 10 you get 0/0 ( which is called an indeterminate form) and that makes no sense to you.

So you graph the function and it looks ‘okay’ around t = 10. Cantor’s work gives us a rigorous grounding for dealing with such situations mathematically. In other words: NO HAND WAVING. We get solid, dependable analytic techniques.

(In this case 0/0 does not come out to be 1 although that some times does happen. Graph the function and see!! Be careful with the brackets/parentheses one mistake and it all turn out wrong

Three times the sine of the quantity10 – t and all of that divided by 10 – t

Mapou: Once we remain in the province of analysis and algebra on variables, calculus is continuous, though you are quite correct that on moving to the world of finite computation, things have gone discrete. And the parallels between difference and differential equations should give pause. However, there is in fact an analytical bridge between Laplace and Z transforms that brings in dynamical elements when physically instantiated; where of course the famed imaginary quantity, sqrt(-1), is material to the analysis, which becomes a way to do 2-d vectors algebraically . . . bringing in forcefully, the point that the math analysis applies to an ideal world of “forms” that gives results we find useful to extend to our everyday real world, and raising issue of the the power of conscious contemplative insightful meaningfully reasoning mind vs blind mechanical GIGO-driven computation. The old fashioned way of saying much of that is to say we do digital stuff using analogue components. The effect of a power supply glitch on a supposedly discrete state system is a classic example. (And I still sometimes reflect on the short-cut that worked, using a silvered mica cap instead of the usual ceramic for power supply decoupling for such a case, and yes the former worked when the latter failed.) KF

PS: I still swear by RPN logic HP calculators, all the way from a fondly remembered HP 21 to my current HP 50, with a couple of HP 48 emulators snuck in there. And, I have a very good report on a Victory, Hecho en la Chine knockoff on the HP 12, for my other set of hats. IIRC the Russians long used their own RPN logic calculator family (Elektronika MK-61 and kin) as last ditch backup for spacecraft navigation.

PPS: Design objectors, you need to recall that Mapou comes down pretty much on your side of the divide on brain-mind issues, if you pardon a few approximations.

PPPS: Anyone who imagines that design thinkers are in lock-step indoctrination should now be seriously re-thinking such crude caricatures.

Jerad, nope, it’s L’Hopital’s Rule. And while Cantor gives one way to look at it, there’s more than one way to skin that cat-fish analytically. And “hand-waving” is dismissive rhetoric. I’d say, that we have various approaches that answer reasonably to the matter, and that it would be helpful to take a look at the implications of Godel’s findings, for schemes that imagine Mathematical, axiomatic systems as a whole to have anything like utter certainty. KF

PS: And, do I need to point to self-evident truths . . . which seem to give vapours to many objectors to design thought?

Mapou, you have hit on one of the interesting points in physics, where a point particle shows itself an ideal, and gives warning on limits. I’d actually say, before we get to infinite self-collapse, we get to everything being a tiny black hole as escape velocity hits c. Where the space-time fabric rips and we are in a different world than we think. Then think about say an electron with its charge, and ask why it does not fly apart under mutual repulsion, as we shrink the presumed ball ever smaller. Oh yes, that does point to the issue that pushing to the all but zero is a mathematical, idealised exercise . . . which has been my point. Classical results here point beyond themselves. Presto, the need for something shaped pretty much like quantum theory and Relativity, with all sorts of interesting paradoxes and unresolved issues. If Godel has mathematicians walking by faith in what they do not see, wait till the physicists come in the door. As in, we all walk by faith and not by sight, the question is in what, why. KF

PS: How do you tell the physicists? Wait till it gets dark, we are the ones that glow faintly blue-green.

KF #204

But Cantor put things like L’Hospital’s Rulr on a firmer foundation.

My use of the term ‘hand waving’ was in response to a previous commentor who used it first!! The accusation started on the other side. I’m just trying to help by giving an example that hadn’t been mentioned.

Jerad, define firmer in a post Godel world, with paradoxes of the infinite lining up to come in the front door. And, I am not so sure about firmer footings when we have now got a context for seeing why infinitesimals worked so well to get Calculus going. But, of course, I see no reason to dismiss Cantor’s work on the transfinite. Where, I insist, transfinite is a useful term that does not carry the baggage that infinite does. Indeed, it allows us to give some substance. But every time I see those triple dots, I see how we say, we extend to infinity with the eye of faith as we cannot actually deliver the infinite in steps. We are only, ever laying out a logical, in-principle case. Somewhere out there the ghost of Plato is laughing uncontrollably and gasping out that he told us so about forms long since. KF

PS: Think about the challenge that non-being, a genuine nothing can have no causal powers; so if there ever was nothing — the ultimate zero — there ever would be utterly nothing, but here we are. Then, think about a suggested infinite succession to the present of causal steps and entities. Then, contrast a contingent cosmos joined to an underlying necessary being. And, ponder the even deeper mystery, eternity. The logic involved is pointing to some pretty deep waters.

KF #207

I’ll just stick with the math I think. I don’t grok an ultimate zero. And I’m not a Platonist, i don’t believe in ideal forms or states.

Godel said we can’t discover everything that’s true about a system by merely piling up theorems on top of axioms. Sometime you have to step out side the system and work back down. But, by then, Cantor’s work was firmly esconsed in the structure of modern mathematics. There is no reason to re-evaluate that which has been previously discovered and built upon. Math isn’t like physics, the old stuff still works just as well as it ever did. That’s why math has theorems instead of theories. Maybe theorems are as close as we can get to ideals eh?

When you say transfinite I’ll just think infinite.

Jerad: I take it you are aware that we are here dealing with an ideal conceptual world that we view as holding a reality that is powerful enough to shape and influence how we analyse and understand the world. Plato’s ghost is laughing, laughing uncontrollably as he looks on at materialists grasping with a world of mathematical and logical realities that point to overturning their preferred vision of reality. Next, Godel’s work had two sides in this context, that not only for rich domains, are there truths unreachable from any given set of coherent axioms but that there is no constructive process to create a set of even admittedly limited axioms that are known ahead of time to be coherent. So, we must recognise the now almost 100 years long state of play with mathematics, which yes has theorems but in a revolutionised context post Godel. And, I do not reject Cantor’s work, I just think we need to appreciate the difference between an ideal world of concepts and the physical one, and how extensions will break down. I remember pondering on the gravity of a point mass in 6th form physics, and also point masses with pendulums. That’s a gateway to seeing the point. Mapou has a point, though of course I have some most profound differences of opinion and views with him. KF

KF #209

I see Godel pointing out something about how the wall has been and can be built. I’m not a research mathematician so I don’t see his work affecting my use of theorems and techniques already established or, in the case of theorems, proven.

We do see some aspects of reality much differently. Which is fine. I just can’t see down your path and mine is sad and limited to you. But there’s enough overlap in our ‘worlds’ that we can agree on much actually. And that’s good. In some ways you do have more hope than me but I find my world view satisfying, exciting and to be savoured. I too see a beautiful world despite all the sham, pain, waste and broken dreams. It’s all part of the whole. And I’m trying to see the whole from a different perspective than you. But we are, kind of, trying to head to similar destinations. We don’t have to walk in each other’s footsteps to appreciate the other person’s journey.

Jerad:

SEP, on the Godel incompleteness theorems:

In short, we have a situation where mathematical proofs have a massive, pivotal proviso, that those of us who use Math should be aware of.

Yes, even Mathematicians walk by faith and not by signt, in the end.

KF

Bob O’H:

What a joke. As if those words require special mathematical meaning.

Please tell us how you came up with that spewage.

Look, if set A has all of the elements of set B AND it has elements that set B doesn’t have, then set A has to have more elements than set B. That is if we can subtract set B from set A and get another set, then set A has to have more elements than set B.

Earth to keith s- If infinity is a journey then there has to be numbers driving that journey.

JWTRuthINLove writes:

How is that changing the rules? Are you saying that we do not use the relationship I posted to determine if one set is a proper subset of another? Do you not understand English?

sergmendes:

It’s called a dictionary and words mean what they do in all scenarios.

Questions my detractors won’t answer:

Why do people think that Cantor is an infallible god?

What good or what use is it to say that all countable and infinite sets have the same cardinality?

What would happen if my claims are correct and Cantor is wrong? (besides math books being changed)

And AGAIN:

From my understanding infinity isn’t a number, it is a journey. Now if you are on a train (Einstein’s train) that had two counters, counter A counted every second of the journey and counter B which counted every other second of the journey. Counter A would represent the set of positive integers and Counter B would represent the set of all positive even integers.At every second throughout the journey Counter A would have a higher count than Counter B, meaning set A would always have more elements than set B.

What my detractors seem to be saying is that counter B should count double every time it counts to allow it to be the same as counter A and because of that they are equal at every other second, and that is the only time you are allowed to look.If my detractors cannot handle that then it is obvious they don’t know jack about infinity.

KF said,

we extend to infinity with the eye of faith as we cannot actually deliver the infinite in steps. We are only, ever laying out a logical, in-principle case. Somewhere out there the ghost of Plato is laughing uncontrollably and gasping out that he told us so about forms long since.

I say.

Amen and exactly!!!

Jerad says,

In some ways you do have more hope than me but I find my world view satisfying, exciting and to be savoured.

I say,

Your world view is only satisfying if you ignore all those pesky edges. The problem is all the interesting stuff is found at the edges and the edges have a way of intruding on you when you least expect it like when you are looking at animated line graphs 😉

Folks chained in the cave do a pretty good job of ignoring what is going on outside most of the time but outside is where all the action is. Plus there are all those nagging questions about where those cool dancing shadows come from in the first place.

Remember when I told you you would not follow the evidence if it meant giving up what you hold dear. This is exactly what I meant.

peace

Mapou,

Do the physicists need to do anything differently, then? If they are already using discrete methods, is there a problem?

Now I understand that a quantum theory of gravity is not here yet, so that presumably is a problem you think needs solving.

But if your ideas were integrated into physics, what specific changes would need to be made? If, as you say, time and distance are illusions, what fundamental quantities would remain? We still make use of clocks and measuring tapes, so they must measure

something, right?Maybe a specific example would be helpful. Suppose you drop a 1 kg stone off a 10 meter building. If time and distance are illusory, what actually happens in your view? Can you describe it mathematically? Would you still use F = ma?

Mapou:Physicists do not use smooth manifolds even if they think they do. They use a discrete and finite computer to do their calculations.You are confusing physics and engineering. Physicists have been using smooth manifolds since Newton and before. While computers have become increasingly important, solving equations, including those that involve a continuum, was how much of physics was done in the past, and is still done. Indeed, even a Euclidean line is infinite in extent, and a Euclidean line segment is infinitely divisible.

fifthmonarchyman @ 219

Philosophy is very good subject for killing time – make up something and talk about it as if it is profound. The philosophy of philosophy is to make sure that whatever is made out to be philosophical should be of no practical use.

Z, once we move beyond analysis and algebra and correlate variables to the real world, we do have to engage finite round-offs, error propagation in such, statistical sampling issues etc. Mapou is right to highlight that in our calcs and measurements when we touch down in the real world, we do go to things that are inherently discrete state and non-continuous. Those triple dot ellipses again. Just, we try to manage our round-offs very carefully to avoid things popping up that bite us really badly. For Newton, his original value on earth-moon forces was about 10% as rounded off and in light of available measures. KF

PS: Was it the old trick to take a calculator and feed in 0.5 then take sqrt or the like over and over again say ten times, then do the inverse and see if you get back to 0.5?

MT if you think phil is irrelevant to the real world, you are dismissing logic, assessment of knowledge, what there is to be known, and more. Yes, hard questions have no easy answers, but we had better ask them and appreciate our limitations, if we are not to find ourselves int eh snares of ideologies that cannot stand the clear light of day outside the cave. I suggest to you, that Plato’s parable on false enlightenment as an embedded consciousness has much to teach us all. KF

kairosfocus:Mapou is right to highlight that in our calcs and measurements when we touch down in the real world, we do go to things that are inherently discrete state and non-continuous.That’s not his claim. Mathematicians and scientists work with the continuum and infinity all the time. Models are never complete or perfectly accurate, but the mathematics of the continuum and infinity have been very powerful tools in mathematics and science.

kairosfocus:For Newton, his original value on earth-moon forces was about 10% as rounded off and in light of available measures.When you calculate the trajectory of a falling object, per daveS’s comment, we reasonably assume the force acts continuously on the object, and that the object moves continuously through space. We can then solve the differential using mathematics based on the continuum. To claim that Euclid, Newton, and Einstein were engaging in “crackpottery”, as Mapou does, is not supportable.

KF @ 225

Logic,assessment of knowledge is not exclusive domain of philosophy. Maths and science are more logical and require applying logic in practical problem solving, so Philosophy is good mostly for shooting the breeze.

Me_Think:so Philosophy is good mostly for shooting the breeze.A couple of millennia ago or so, philosophy was quite refreshing.

me_think says,

The philosophy of philosophy is to make sure that whatever is made out to be philosophical should be of no practical use.

Zack says,

Mathematicians and scientists work with the continuum and infinity all the time.

I say,

nuff said

peace

fifthmonarchyman:nuff saidYou do understand that the continuum is fundamental to Euclidean geometry and to Newtonian mechanics, both of which have had huge practical benefits?

MT, Phil is the home discipline for both logic and theory of knowledge, aka epistemology,indeed both are main branches. KF

zac asks

You do understand that the continuum is fundamental to Euclidean geometry and to Newtonian mechanics, both of which have had huge practical benefits?

I say,

Of course I do.

I wonder if you and Me_think know this also?

peace

Z: I repeat, M has a point, once we move to the real world of observation, measurement and comparison with expected outcomes, the discrete state comes right in the door. It may not be all that he meant to say — and I have explicitly not endorsed all he says, and indeed have strong differences on ever so many subjects — but it is a point that we all need to reckon with. KF

PS: Whoever pointed out above how an inverse square law would act as the radial distance shrinks towards zero also has made a point we should also attend to.

kairosfocus @ 231

Yes. So ?

KF at 233:

But clearly no one is disputing this, so I’m not sure why it bears repeating?

fifthmonarchyman:Of course I do.Then you will reject Mapou’s contention that infinity is “crackpottery” and the continuum “nonsense”.

kairosfocus:I repeat, M has a point …His contention was that infinity is “crackpottery” and the continuum “nonsense”.

Zac says,

Then you will reject Mapou’s contention that infinity is “crackpottery” and the continuum “nonsense”.

I say

I already did so way back at 113

I wonder if you will reject Me_think’s contention that the Philosophical is of no practical use

peace

fifthmonarchyman @ 237

Can you give 2 major

exclusiveuse of philosophy ? one is killing time and the other ?me_think asks

Can you give 2 major exclusive use of philosophy ? one is killing time and the other ?

I respond

How about describing concepts like infinity and the continuum that are “fundamental to Euclidean geometry and to Newtonian mechanics”.

will that work for you?

peace

fifthmonarchyman(113):It looks to me that Joe and Mapou are all wet on this one.Sorry. Our spam-filter had grayed out the comment, probably because of the meta-discussion.

fifthmonarchyman:I wonder if you will reject Me_think’s contention that the Philosophical is of no practical useAs we said, a couple of millennia ago or so, philosophy was quite refreshing.

fifthmonarchyman:How about describing concepts like infinity and the continuum that are “fundamental to Euclidean geometry and to Newtonian mechanics”.Philosophers grappled with infinity, but made little progress. Suppose you could say philosophers held the problem at bay, but it was mathematicians, such as Georg Cantor, that resolved the fundamental issues.

Zac,

Since we agree that infinity does not exist in the physical universe how is it that Cantor was not doing philosophy?

fifthmonarchyman @ 239

You really think philosopher’s infinity is equal to scientists infinity? Philosophy is not about finding solutions to problems. Scientist don’t consult philosophers before finding solutions.

Most scientists of bygone era indulged in philosophy too, may be that made you to think philosophy is of practical use.

Z, your view of phil seems to be rather narrow. It is a foundational discipline in its own right and has much to say to other disciplines, including both science and mathematics. Indeed the Godel results as above are regarded as phil results, not just mathematical. And, I am by no means so convinced that Mathematicians have fully uncovered the nature of the infinite or more properly the transfinite. Contributions — indeed important ones — yes, covering the whole story, not at all. KF

Me_think asks

You really think philosopher’s infinity is equal to scientific infinity?

I say,

How exactly is it different.

peace

DS: It is being side-stepped as opposed to disputed. And, the case of the point charge or mass will bear much serious reflection. As, once did the seemingly idle question of taking a ride on a beam of light. Or, comparing a falling apple to the Moon swinging by in orbit. KF

fifthmonarchyman:Since we agree that infinity does not exist in the physical universe how is it that Cantor was not doing philosophy?Two doesn’t exist in the “physical universe” either. (Edited for clarity.)

If you envelope mathematics within philosophy, then it’s philosophy. If you envelope science within philosophy, then it’s philosophy. If you envelope pinochle within philosophy, then it’s philosophy. You may want to define philosophy.

We’re not against philosophy, by the way. A couple of millennia ago or so, philosophy was quite refreshing.

fifthmonarchyman:How exactly is it different.Among other things, there’s more than one cardinality of infinity.

fifthmonarchyman @ 245

Do you see any thing in there which leads you to ‘divergence’ ? (which is mathematical calculated infinity)

zac says,

You may want to define philosophy.

I say,

Maybe you might want to define philosophy.

I believe it is people like you and Me-think that want to segregate and exclude philosophy from life because it supposedly does not deal in the testable ie phyiscal.

For me knowledge is knowledge no matter how it is acquired.

me_think says,

Do you see any thing in there which leads you to ‘divergence’ ?

I say

No, infinity is infinity no mater if we are talking about God or the total number of primes

Me_ think asks,

(which is mathematical calculated infinity)

I say,

Which is not “mathematical calculated infinity”?

Peace

fifthmonarchyman:Maybe you might want to define philosophy.We use the traditional definition, “the study of ideas about knowledge, truth, the nature and meaning of life, etc.”

fifthmonarchyman:I believe it is people like you and Me-think that want to segregate and exclude philosophy from life because it supposedly does not deal in the testable ie phyiscal.Not sure why you would say that. For instance, ethics is an area of philosophy, and is intimately connected to the human condition.

fifthmonarchyman:No, infinity is infinity no mater …Well, it turns out that not all infinities are the same cardinality. Some infinities are bigger than others, much much bigger.

Zac says,

Well, it turns out that not all infinities are the same cardinality. Some infinities are bigger than others, much much bigger.

I say,

Could not agree more. I never said that all infinities are the same cardinality.

zac says,

We use the traditional definition, “the study of ideas about knowledge, truth, the nature and meaning of life, etc.”

I say,

The study of infinity fits in there quite nicely thank you very much. It always has.

why would it not?

peace

fifthmonarchyman:I never said that all infinities are the same cardinality.No, but philosophical meandering didn’t make any headway on the problem.

fifthmonarchyman:The study of infinity fits in there quite nicely thank you very much.Cantor subsumed it under mathematics.

Zac says,

Cantor subsumed it under mathematics.

I say,

Mathematics when it deals in infinities fits in there rather nicely as well. Always has. Why would it not?

peace

fifthmonarchyman:Mathematics when it deals in infinities fits in there rather nicely as well. Always has.What is the distinction between infinity-ness and two-ness in terms of philosophy? What does philosophy have to say about infinity-ness or two-ness that is not subsumed within a mathematical framework?

daveS:

Measurement does not change. It is automatically constrained by the discreteness of reality. We cannot go past the discrete Planck scale. What changes is our understanding of the universe. As I said earlier, the idea of a continuous universe is a disastrous mindset because it prevents us from seeing nature as it is. Once you accept that the universe is discrete, it opens up a whole new vista on reality, a view that was heretofore hidden to us. You immediately realize that distance (space) is abstract, a perceptual illusion: it does not exist. Only particles and their properties exist. The idea that we are moving in space (a la Newton) or spacetime (a la Einstein) is 100% hogwash in the not even wrong category.

Once you realize that there is no distance, then it is obvious that nonlocality is the rule, not the exception. Einstein’s “spooky action at a distance” objection to QM can be immediately dismissed as the product of ignorance.

We can expand on this further. If distance/space is an illusion, then position is not a property of space but a property of the particles themselves. And if position is an intrinsic property of particles, then they are obviously absolute and the idea that position and motion are relative is just bogus nonsense, pure crackpottery of the worst kind. And if position is an intrinsic variable property of particles, there is no reason that they must only change from one value to an adjacent value. In the not too distant future, we will have technologies that will allow us to instantly move from anywhere to anywhere.

It goes much further. If distance does not exist, motion consists of just changing the positional properties of particles. Every change is an effect that requires a cause. The idea that inertial motion requires no cause is thus stupid and dumb. It follows that we are moving in an immense lattice of energetic particles without which there could be no motion. One day, we will learn how to tap into the lattice for energy production and propulsion. We will have unlimited energy production and vehicles that will travel at tremendous speeds without any visible means of propulsion. I foresee an amazing time of free unlimited energy, floating sky cities impervious to weather, earthquakes, tsunamis and other natural disasters, New York to Beijing in minutes, Earth to Mars in hours, billions of robots zipping around the earth and the solar system, etc. This is our future.

I’ve written about this stuff on my blog. Here are a few links:

How Einstein Shot Physics in the Foot

Why Einstein’s Physics Is Crap

Physics: The Problem with Motion

PS. I will not go into gravity because it assumes that we exist in a 4-dimensional lattice and that we are moving in the fourth dimension at c. Gravity is caused by our interactions with the lattice: it is nature correcting violations to the conservation of energy. It gets a little complicated and this forum is not the place for it.

Zac says,

What is the distinction between infinity-ness and two-ness in terms of philosophy?

I say,

Both 2 and infinity are abstract platonic forms they just represent different quantities.

When you are talking about abstract “two-ness” you are already deep in philosophical territory.

You say,

What does philosophy have to say about infinity-ness or two-ness that is not subsumed within a mathematical framework?

I say,

what?

How is the “mathematical framework” different than from the “philosophical framework”?

It seems to me that all mathematics at least that beyond simple concrete arithmetic is part of the “philosophical framework”?

Please describe exactly where mathematics ends and philosophy begins in your worldview?

peace

fifthmonarchyman:Both 2 and infinity are abstract platonic forms they just represent different quantities.A couple of millennia ago or so, philosophy was quite refreshing.

fifthmonarchyman:It seems to me that all mathematics at least that beyond simple concrete arithmetic is part of the “philosophical framework”?If you subsume mathematics within philosophy, then it’s philosophy. If you subsume science within philosophy, then it’s philosophy. If you subsume pinochle within philosophy, then it’s philosophy. You may want to define what you mean by philosophy.

Mapou,

Ok, thanks. I will check those links when I get back online in a few days.

Zac says,

You may want to define what you mean by philosophy.

I say,

“the study of ideas about knowledge, truth, the nature and meaning of life, etc.”

Philosophy is simply the love of wisdom.

At some point materialists like yourself have decided that you don’t care about “wisdom” but are only concerned about the purely practical and any thinking beyond that was seen to be old fashioned fruitless nonsense.

The problem is as we see with Cantor and infinity, philosophical wisdom keeps intruding on the practical and everyday.

In the cave it’s the edges are where all the action is and the edges keep showing up in the darnedest places

Merry Christmas

quote:

Wisdom cries aloud in the street, in the markets she raises her voice; at the head of the noisy streets she cries out; at the entrance of the city gates she speaks: “How long, O simple ones, will you love being simple? How long will scoffers delight in their scoffing and fools hate knowledge?

(Pro 1:20-22)

end quote”

peace

fifthmonarchyman:At some point materialists like yourself …We’re not a materialist.

We asked you how philosophy informs us about infinity-ness and two-ness, and you referred to a philosopher from 2400 years ago. Our response was and is, a couple of millennia ago or so, philosophy was quite refreshing.

@Joe (215):

A “contrived relationship” is irrelevant for set theory. You’ve extended the set theory with the “contrived relationship” specification.

I’m an engineer. It’s not about understanding English, but about using a domain-specific glossary.

Joe #217:

All you had to do was ask.

I don’t know of anyone who does.

It’s an important and counterintuitive advance over the idea that they have different cardinalities.

Your claims lead to inconsistencies. Mathematicians prefer consistency to inconsistency, for obvious reasons. ‘Joe Math’ is inconsistent and therefore stillborn.

Joe #218:

If you insist on using that metaphor, then infinity is the

entirejourney. Your mistake is that you keep referring to finite portions of the infinite journey as if they were infinite.I was wondering when ‘choo-choo math’ would make its appearance. But if you’re simply counting elapsed time in seconds, what’s the point of the moving train?

Here’s a crucial point, Joe, so please pay attention: At any

finitepoint in time, counter A would represent thefinitenumber of seconds that had elapsed, while counter B would represent thefinitenumber of two-second intervals that had elapsed. Finite. Finite. Finite. Not infinite.After the first second, yes.

No. You are thinking of sets A and B as

finitesets that are continually growing.They are not finite. They are infinite. They are not growing toward infinity. They already are infinite.No, what your detractors are saying is that infinity is not a number. After T=1, there will never be another point in time when counter A is equal to counter B.

Your mistake is to think that counter A and counter B are both counting up to some humongous number called ‘infinity’. They aren’t. Infinitity is not a number.

When we say that your two sets, the positive integers and the positive even integers, have the same cardinality, we are saying that they can be placed into a one-to-one correspondence.

We are not saying that your two counters will converge on the same number.Infinity is not a number, Joe.What good or what use is it to say that all countable and infinite sets have the same cardinality?

What’s the importance? If we listen to Occam counterintuitive is most likely wrong.

What would happen if my claims are correct and Cantor is wrong? (besides math books being changed)That you manufacture. However t is inconsistent to use one methodology to determine if one set is a proper subset of another and then contrive a methodology to see if they have the same cardinality.

It is inconsistent to say that one set as all of the elements of another PLUS elements the other doesn’t have and then say they have the same number of elements.

Not that keith will understand that rebuttal of his “answers”.

Zac says,

a couple of millennia ago or so, philosophy was quite refreshing.

I say

check it out

quote:

Barfield never made me an Anthroposophist, but his counterattacks destroyed forever two elements in my own thought. In the first place he made short work of what I have called my “chronological snobbery,” the uncritical acceptance of the intellectual climate common to our own age and the assumption that whatever has gone out of date is on that account discredited.

You must find why it went out of date. Was it ever refuted (and if so by whom, where, and how conclusively) or did it merely die away as fashions do? If the latter, this tells us nothing about its truth or falsehood. From seeing this, one passes to the realization that our own age is also “a period,” and certainly has, like all periods, its own characteristic illusions.

end quote:

CS Lewis

peace

From my understanding infinity isn’t a number, it is a journey.Why is it a metaphor? And the journey never ends so there isn’t any “entire journey”.

Your mistake is thinking that is what I do. But then again you love to hump a strawman.

So children like you could play too.

At EVERY finite point in time, forever. Forever. Forever. Forever- infinity even.

Well that is YOUR mistake.

The sets are growing forever. Infinite growth.

Right, it’s a journey. Infinity isn’t something that “just is”. It doesn’t just appear

I never said nor thought you said that. My argument doesn’t require it.

Zacky-O:

Pure poppycock. There is no such thing as a mathematics of the continuum and infinity. If it uses numbers, which are discrete by definition, it is discrete, period. Jumping up and down, foaming at the mouth and screaming that your math uses continuity and infinity makes no difference other than you putting your foot in your mouth. And since you insist on referring to yourself as a “we” (are you possessed by demons, goddamnit?), you all have your feet firmly planted in your mouths. LOL.

You can assume anything you want about bodies moving continuously through space but it’s still complete BS.

They were all crackpots in the things they did not understand. Pathetic crackpots, in fact. I revere only the truth, not some mortal beings who are long dead and buried. If they were so hot and mighty, where are they? Humans die but the truth remains.

JWTruthInLove:

Only for this one obviously irrelevant aspect of it. You cannot contrive a relationship to say that one set is a proper subset of another. Do you understand that?

{apples, beans} is not a proper subset of {a, b, c} just cuz I can contrive a relationship using only the first letters of the words in the first set.

So I would say the word is very relevant to set theory. Care to try again?

Me_thinks,

You claim that philosophy is useless, and yet you have no counter to my argument about how using Cantors method for pairing objects in two infinite sets is completely arbitrary and therefore solves nothing. Once I change the names of different kinds of numbers (rational, real, whole, integers, etc…) to one name, which includes any kind of number, Cantors arguments becomes meaningless. If I no longer have names for all the different kinds of numbers, then one infinite set simply contains varying numbers, and another infinite set also contains varying numbers.

I have no reason to say one set is bigger than another, if I have no separate names for different types of numbers, they are all just numbers. Why would I use his method of bijection, if the names for numbers is just numbers, rather than some man made category for different kinds( There are lots of other ways of categorizing numbers if I insisted on giving them the name one wants, why didn’t he match numbers with zeros to numbers without zeros, its just as arbitrary).

Cantor’s argument thus fails. Philosophy beats math. You, and Zacheriel, and Jerad have no defense for that.

Joe:

Correct. Why is it that some people refuse to accept this simple truth?

Claiming that infinity exists is like claiming that there is such a thing as an infinitely smooth circle even though Pi can never be fully computed. Never. The BS we are taught in science is enough to make a grown man cry.

I see that keiths is back with his subjective morality once again telling the rest of us how we OUGHT or OUGHT NOT behave.

Feel free to join me in laughing at him with a very merry HO HO HO!

phoodoo @ 269

You don’t need M&M’s or ‘named’ numbers. Take set of binary digits – cantor’s diagonal argument will still work.

Folks:

First, Merry Christmas.

Next, I think we should (–> and yes, there is that pesky OUGHT again; here, duty of care to accuracy, balanced reasonableness and fairness) distinguish the world of abstract concepts and sets, functions and continuous lines or variables, and our finite, discretely observed, measured and computed world.

It is reasonable to apply mathematical models to the world, but we need to be aware of limitations. And indeed the point that we speak of point particles with masses and force fields of inverse square character should serve as a simple first warning on limitations.

(Cf the exchanges here on ways others have pondered this matter, and its enduring subtle significance. Including a pointer to the idea of strings. Mathematical models are not going to be exact physical realities, especially where there is some simplification involved, and/or where elaboration in an educational context will run into deep waters very fast. [This opens up the problem of trusting authorities with simplifications of complex matters, which has become a sore point in a context where crucial areas of science are too often increasingly tainted with ideological agendas. In such cases, appeal to consensus on the one hand may not be good enough, and on the other, we should not get into such a suspicious mindset that we dismiss on a blanket basis without hearing out the deeper explanation or context. In my own experience, I found that telling students that there was a simplification involved [–> E.g. sun and planets atomic model backed up by Rutherford’s experimental findings] and suggesting look-up for those who were going on further, was often enough. Empirically reliable enough models used within zones of tested reliability have their own justification [–> there’s that pesky OUGHT again], but let us not fool ourselves [–> and again] that we have seen all there is to see. Even so “simple” a thing as a mirror implies on laws of reflection a virtual half-cosmos behind it.])

I now suggest that we are finite, fallible, bounded in our rationality, and too often ill-willed.

In that context, we can pause and then recognise that whatever we actually do is bound by our finite space-time existence, and that actual calculation resulting in actual numerical values will indeed always be discrete.

It is algebra and analysis and onwards that allow us to think in terms of a mathematically precise meaning of continuum and the transfinite, as well as the complex domain, vector spaces and more, much more.

And yes, we have now left the concrete, empirical world.

We are in a world of mathematical, abstract conceptual objects that nevertheless have such realities governed by the logic of coherence and systems of articulation that we ever so often find astonishing predictions based on the mathematics.

Let us be grateful for that, and let us ponder that if mental abstractions work so well in the empirical world, maybe they were in fact built in from the outset, i.e. the conceptual-logical world of mathematical forms may well be part of the design architecture of the observed, experienced world.

And of course that means we are back at the observation that the ghost of Plato is laughing, he had a point. Though not necessarily the whole story, as usual.

Where, the continuum does point to the infinitesimal, and so also to the transfinite.

(To my mind, maybe we should go back to thinking in terms of looking at the hyper-reals and their reciprocals, the infinitesimals, as a gateway into Calculus, and onwards wider analysis. I see no reason to ignore a model that has been continually cropping up for over 2,000 years and has been yielding pretty good results. And indeed looking back at old editions of Granville Smith and Longley, I found the use of infinitesimals quite sensible as an introduction back when. The approach of limits, neighbourhoods and the like, is a bit clumsy at that first level, by contrast. And, as basically a physicist, I see no good reason why we should exile time from consideration as we look at space. It is after all x(t), y(t) etc that we really deal with! Trajectory is not suspect, and when we say that the finite experiential process allows us only to point to the infinite that is a way of saying that we have jumped to the abstract world, and why not call them forms and be done?)

Plato’s parable of the cave — 2400 years on, pace dismissiveness to phil — still has bite, even in terms of the difference between experienced worlds of mundane existence, and the abstract space of idealised concepts.

Which can have their own surprising existential powers.

I find it intriguing that one cannot fold a paper clip into a square circle, because the core properties of the two elements stand in mutual contradiction.

That is an astonishing point, if one thinks about it.

A logical constraint can lock out realities that we might otherwise think possible. (And is it just me who notices how many waste paper baskets are done as circles transitioned to squares or vice versa? Or more properly, approximations . . . )

Our world is full of clues that point beyond itself.

An entirely appropriate point on a day when we celebrate one whose birth was heralded by “Angels we have heard on high . . . ”

Merry, thoughtful, grace-ful Christ-mas.

KF

Grace and Peace to you KF!

And, the same to you. De beaches are nice, much nicer dan ice . . .

(But, how often have I parked right next to a beach, only to be heading in the opposite direction to work. Not to wade, swim or fish . . . and there’s a new year resolution for you!)

KF

Calculus as invented by Newton and Liebniz works because of the continuous nature of mathematical functions. For a given modelled real life situation the math of continuity works and gives us answers that can be measured on the ground.

If it was all just hogwash it would have been dropped decades if not centuries ago. It is some of the powerful,useful and dare I say beautiful analytic tools invented by human beings. And it works. It helps you solve real world problems. And it rests on continuity.

Take a class in complex analysis (that’s calculus with imaginary numbers). Talk about falling down the rabbit’s hole! And there are good, solid, real world applications. I couldn’t believe it but it works. And it’s used. By engineers every day.

Fourier analysis is an amazing and powerful technique, again based on continuous functions. I’ve had engineers tell me, whatever else you teach make sure they understand what goes into Fourier analysis. It works.

The mathematicians who developed Analysis (the area of math that includes Calculus) were generally working on physical problems. And they had (and continually have) other mathematicians scrutinise and criticise their work. Even as an undergraduate no holds are barred. You get something wrong, you’re told. Unless you’ve been through it it’s hard to convey the impossibility of getting something accepted that isn’t true or doesn’t work.

Mung:

I am?

Merry Christmas, Mung. I hope Santa brings you less confusion this year.

keiths:

Merry Christmas keiths.

Why OUGHT Santa bring me anything?

Why OUGHT Santa bring me “less confusion”?

Wny OUGHT anyone take your subjective morality seriously?

Why is it that in your subjective moral announcements you presume that everyone OUGHT TO agree with your subjective moral announcement, as if it were an objective truth?

Jerad @276:

Amazing. Nobody here said that calculus was hogwash. You are blatantly lying in order to make a stupid point and you know why, Jerad? It’s because you are a jackass. If calculus used continuity as you claim, why is it that you cannot use it to create a perfect circle, huh? Show us your infinitely smooth circle or shut the hell up.

Show us a reason that we should continue to argue with jackasses.

Jerad:

I use Fast Fourier Transforms (FFT) in my work in speech recognition and I can assure you there is nothing continuous about it. It’s 100% discrete.

Mapou #279

The parametric equation:

x = sin(t)

y = cos(t)

0 </= t </= 2pi

Gives you a continuous circle. And it's differentiable.

e^(it) gives you a circle in the complex plane.

y = sqrt(r^2 – x^2). Gives you the top half of a circle radius r.

All of these are continuous functions. You can zoom in as far as you like and the curves stay smooth.

In polar coordinates it's dead easy: r = whatever radius you want. And you can also do calculus with that.

You don't have to argue with me at all. I'm just telling you they way it is. Most of the theorems of calculus depend on continuous functions. If the theorems weren't true then calculus would be on pretty shaky ground. You can link them up if you don't believe me.

Mapou #280

Sine and cosine functions are continuous functions. Is that not what you use in FFT?

What do the theorems behind FFT say?

Folks,

it’s Christmas Day, let’s ease back several notches. In addition, I suggest the only place where we have perfect circles is an ideal world of mathematical forms. Once we hit concrete reality, we are dealing with messy approximations and round-offs. And, I am glad to see strong affirmation of inherently mental realities that speak with power into our material world.

Here is Plato’s ghost, laughing as he opened the door for theism by way of perceiving design . . . ironically, an act of mind:, The Laws, Bk X

Ho, ho, ho, Merry Christmas.

KF

KF #283

Sounds good to me! It’s 8:30 Christmas morning in England.

MERRY CHRISTMAS!!!!!!

~jerad

Thank you, kairosfocus. Merry Christmas to you.

KF @ 283

Perfect example of shooting the breeze.

Merry Christmas to all.

MT: Oh, the irony of your handle! The self-moved agent is the only entity truly free to think for itself. And, an infinite stepwise successive causal chain is an attempted traversal of a countably transfinite set, which fails. Enjoy the day. KF

Jerad, de beaches are nice, much nicer dan ice. Even so, enjoy the day. KF

PS: That’s a Jamaican carol I am citing, BTW.

Joe wrote:

So…

In set theory, “power” is the ability of a set to act on another set.

“Fuzzy sets” are ones covered in bristles.

“Trees” are plants too large to grow in a “morass”.

A “club set” is a gig in a private dance-hall.

“Large cardinals” are overweight catholic patriarchs.

“Scott’s trick” is a way to use a transporter when travelling at warp speed.

A “union” is a number of sets that gather together to engage in collective bargaining.

And of course “set theory” itself is a hunch about where badgers live.

Roy

All,

Just for fun here is some more cool “shooting the breeze”.

You just never know when the edges will show up.

http://nautil.us/issue/13/symm.....nrose-tile

Peace (on earth)

KF @ 287

from the article:

quote:

the placement of one Penrose tile can affect things thousands of tiles away—local constraints create global constraints. But if a crystal forms atom by atom, there should be no natural law that would allow for the kind of restrictions inherent to Penrose tiles.

It turns out crystals don’t always form atom-by-atom. “In very complex intermetallic compounds, the units are huge. It’s not local,” says Shechtman. When large chunks of crystal form at once, rather than through gradual atom accretion, atoms that are far apart can affect one another’s position, exactly as do Penrose tiles.

end quote:

I wonder if anything else in nature forms all at once rather than through gradual algorithmic steps?

😉

peace (on earth)

fifthmonarchyman @ 290

It is far more useful to study Quasicrystals

LoL!@ Roy- All of those words have meanings- accepted and standard meanings- in set theory. OTOH contrive and derive remain the same and I have explained it all. Strange that people ignore my explanations and prattle on as if their blatherings mean something.

To review

we went from shooting the breeze about infinite non-repeating tiles to naturally occurring quasicrystals to squishy practical self-assembling organics at the center of life in less than 40 years

http://www.wired.com/2014/03/organic-quasicrystal/

There is a whole world waiting to be explored just outside the cave. It be a shame to miss it

me+thinks,

I have no idea what you are saying. How does the concept of binary digits help with the problem of a totally subjective method for creating bijections of two sets?

I can two sets and think of thousands of ways to match one from one set up with a counterpart in the other. The problem still doesn’t go away. If the premise at the beginning is that each set contains an unlimited amount of elements in the sets, then neither set can be said to contain more or less than the other, they can’t both contain the same amount of elements and different amount of elements at the same time-math can not rescue the logic out of the problem.

The diagonal line game is a canard, that traps people who refuse to think about it very deeply, but instead prefer to believe whatever they are told.

fifthmonarchyman @ 295

Is non-repeating tiles philosophy ?

Me-think says,

Is non-repeating tiles philosophy ?

I say,

Of course thinking about infinite non-repeating patterns is philosophy. Why would it not be?

The abstract mental plane on which the tiles are arranged does not exist in the phyiscal universe it exists outside the cave.

What you see in the phyiscal universe are merely approximations “shadows”.

peace

Where is philosophy in tiles arrangement on a plane ? It is geometric arrangement. Plane is a geometric structure. It is not ‘outside’ Plato’s cave or dugout.

Me_Think says,

It is geometric arrangement.

I say,

Is an imaginary infinite plain some how part of the phyiscal universe because you attach the word geometric to it? Plato did not think so

http://www.storyofmathematics.com/greek_plato.html

http://www.ucl.ac.uk/~uctymdg/.....re%201.pdf

peace

phoodoo @ 296

It is what Cantor’s diagonal argument is about (refer Cantor’s 1891 article). It is

thediagonal argument using uncountable set.You can ponder this too : If you stand in between 2 mirrors ( thus creating an infinity mirror), and your friend stands between 2 mirrors of smaller length, are the ‘infinities’ created by your mirror images and your friend’s images same ?

me_thinks,

You seem to just being pulling out half concepts, and trying to string together some kind of argument, that I am pretty sure you don’t even know what you mean.

I know what Cantor’s little card game is, but the point is there is no reason to insist on pairing things up the way he does, as if its the only way. Furthermore, its just a silly diagram on a piece of paper, I could place all the numbers with curly shapes on one line, and numbers with straight lines on the other, it is just as meaningful.

And, you do realize that the reflections in mirrors don’t really go one forever, right?

fifthmonarchyman @ 300

Why are you confused ? Plane is geometry, whether you imagine it as part of physical universe or as abstract structure doesn’t matter.

phoodoo @ 302,

No.You are discussing Cantor’s diagonal argument and I am asking you to refer to his argument which is based on infinite sequence of binary digit set.

Pairing up is a simpler way (than counting elements in set) of seeing if sets are equal.

You do realize that fading of images is because of not enough lux reaching your eyes? Technically, the mirror is reflecting at geometric progression of 1/n^2, whatever the nth image may be.

Me_think says

Plane is geometry, whether you imagine it as part of physical universe or as abstract structure doesn’t matter.

I ask.

Are you familiar with the Platonic forms or the Platonic solids?

https://www.youtube.com/watch?v=H9Q6SWcTA9w

https://www.youtube.com/watch?v=gVzu1_12FUc

Peace

fifthmonarchyman @ 305,

No. There are already enough topics to kill time with.

As opposed to a non-continuous circle? The geometry of non-continuous circles must be really tricky…

Me_think says

There are already enough topics to kill time with.

I say,

You might want to check out the videos

Roger Penrose the guy who discovered the infinite non-repeating tiles we are discussing seems to kill a lot of time with these particular topics.

peace

MT: Yes, there were debates back then too. Let’s just use a modern set of terms, reflexivity and lag or memory, forming feedback loops. Or, are you unwilling to acknowledge that (a) we influence our own selves across life and also in much shorter frames to milliseconds, and (b) we are responsibly free and self-acting? If the former, that’s a plain breach of common sense to the point that you imply that one cannot acquire a self-formed view, or learn a profession and act in light of the acquired knowledge and skills. If the latter, you undermine learning, knowing, cognition and even the ability to choose to acknowledge force of evidence and reason. KF

fifthmonarchyman:You must find why it went out of dateNever said it went out of date. It just got stale. With regards to infinity, philosophical musings have been supplanted by mathematical proofs.

Mapou:There is no such thing as a mathematics of the continuum and infinity. If it uses numbers, which are discrete by definition, it is discrete, period.And if it uses continuous functions, then it is continuous by definition.

Zachriel:To claim that Euclid, Newton, and Einstein were engaging in “crackpottery”, as Mapou does, is not supportable.Mapou:They were all crackpots in the things they did not understand. Pathetic crackpots, in fact.Eppur si muove.

phoodoo:If I no longer have names for all the different kinds of numbers, then one infinite set simply contains varying numbers, and another infinite set also contains varying numbers.The naming of sets doesn’t change whether a given set of numbers if countable or not. Take the set {1, 2, 3}, for instance.

phoodoo:I have no reason to say one set is bigger than another, if I have no separate names for different types of numbers, they are all just numbers.The naming of sets doesn’t change whether a given set of numbers if countable or not. Take the set {cat, trout, dolphin}, for instance.

Mapou:Nobody here said that calculus was hogwash.“Calculus has historically been called ‘the calculus of infinitesimals’, or ‘infinitesimal calculus’.”

Mapou:If calculus used continuity as you claim, why is it that you cannot use it to create a perfect circle, huh?See http://mathworld.wolfram.com/Circle.html

phoodoo:I can two sets and think of thousands of ways to match one from one set up with a counterpart in the other.Not in every case. You can’t find a one-to-one correspondence for the natural numbers and the real numbers.

Joe:

Glad you caught this. Interesting how the same math is used for both continuous and noncontinuous circles, eh? Jerad is on a roll.

If one uses rank, ie position in the set, then one should be able to do so.

Mapou, All non-continuous circles are used as shopping cart wheels. 😎

KF:

Actually, this ideal world of mathematical forms in which perfect circles exist is an impossible world. Why? It’s because numbers are discrete by definition. So if anything uses numbers (this includes all mathematical functions), it is discrete. So, no matter how continuously smooth you think your circle is, it isn’t. Surprise!

IOW, if Pi cannot be written down, all possible circles are discrete. It’s not rocket science, really.

Mapou:So if anything uses numbers (this includes all mathematical functions), it is discrete.There are many continuous mathematical functions.

Mapou:if Pi cannot be written down, all possible circles are discrete.Pi.

Demon possessed Zacky-O:

Funny, I have not seen one yet. Calling something continuous does not make it so. I could say that your cranium is infinitesimally small, too, you know. In your case, it might be true. Let’s see. How many demons can fit in an infinitessimally small cranium? LOL.

At least one of your demons has a sense of humor. But sooner or later, the day of unspeakable torment will be upon you all.

The functions I listed are all continuous over their domains. And not just because i said so, because they meet the definition. Look it up.

Sin(x), cos(x), 2^x are continuous for all real values of x. Log(x), sqrt(x) are continuous ober their domains. Again, because they meet the criteria. The Wikipedia page is as good a place to start as any but one way to think about is: you can zoom indefinitely far over the domain and the graph will not get jagged.

Fractals are nowhere continuous. I think. I’ll have to look that up later.

As I said, the definitions and examples can be easily found. Any basic calculus book will have a good discussion of the topic and will explain why continuous functions are so important in calculus.

Zac, zac, zac…

A set of infinite numbers is already uncountable! Since both sets contain un-countabilities, your point has no meaning. So you have just made me realize you don’t understand the problem at all. You are just searching for answers online.

Its kind of disappointing, because I then know that any counter arguments you are proposing are really just loose threads that aren’t based on a real contemplation of the issue. Like when you say there is no one to one correspondence, I could try to draw this point out and ask what you mean by a correspondence, but since you don’t know what it means, I guess we will gain nothing.

You can zoom IN anywhere over the domain.

DUH.

I hate typing on iPhones.

Phoodoo,

A one-to-one correspondence means you can show no element is left unmatched to a member of the other set.

For example: here’s a one-to-one correspondence between the positive evens and the positive odds

1 2

3 4

5 6

Etc

If you specify an element in either set I can tell you what element in the other set it’s marched with. No element is left out it’s a one-to-one correspondence so the sets are the same size.

This is how mathematics uses the terms.

Jerad:

The point is not that you can’t zoom in till you turn green. The point is that, regardless of how far you zoom in, it is still discrete numbers. It’s

always discrete. Deny until you pass out.You know, Parmenides and his disciple Zeno pointed out the stupidity of infinite series ages ago. You bozos still have not learned the lesson. No wonder physics is filled with braindead monstrosities like time travel, infinite parallel universes and other crap. Intellectual incest is alive and kicking in science.

The concept of continuity is one of the biggest lies in science, right next to Darwinian evolution and the time dimension.

@phoodoo:

Mapou

You can keep zooming in without limit, there is no smallest interval. And all numbers can be written with infinite decimal expansions. Some are nice enough not to have a pattern and you can still write some of them down.

One solution to x^2 = 2 is the sqrt(2). Exactly.

The ratio of the circumference of a circle to its diameter is pi. Exactly.

There are an infinity of numbers between any two numbers you give me on the number line. That matches the mathematical definition of continuous.

Mapoi

The concept of continuity has proved very powerful in mathematcs and it has several equivalent dedinitions. You may think it’s a lie but without it some of the theorems supporting FFT would not be true. You can still grind out the calculations based on someone else’s algorithm but it doesn’t mean continuity doesn’t underlie the whole endeavour.

And your dislike doesn’t make it false.

Jerad, I am still waiting for you to explain how something that uses numbers (the epitome of discrete entities) can be non-discrete. And I’m still waiting for you to explain how something that can never be fully counted by definition makes any sense in reality. But then again, forget it. I know you and that demon-possessed ZacKy-O meister are weavers of lies and deceptions. You can always come up with another cockamamie non-answer.

Mapou, if you really find much of mathematics equivalent to voodoo then I guess we’ve little to talk about. I’ve been trying, in my imperfect and limited way, to explain some things but your view has little in common with the foundations and theorems of calculus. I don’t mind but you calling it lies and hogwash is a bit silly really. Still, it’s your opinion. But there’s a couple hundred years or more of mathematicians that disagree with you

You are right that you and I have little to talk about since I love math and I use it all the time in my research. You are a complete bore, Jerad. A jackass.

JWTruthInLove,

The same contradiction which lies at the heart of Cantors proof, also lies in the contradiction of calling an infinite set countable. They are simply two ways of saying the same thing. Once you have told me all of the elements of your “countable infinite set” I can then give you another one you didn’t count.

So its not math, or wikipedia which helps you out of the problem, its the problem of bad definitions giving bad results. Cantor was playing a game that seems to solve a paradox, but the paradox is already built into the question, so the paradox can’t be removed. We can’t halve infinity, or double infinity, so we don’t have a starting or an end. What’s the first countable in an infinite set? There isn’t one.

Take your early geometry lessons as an example. We start with the assumption that we have a point in space which is infinitely small, but discrete. We then have an imaginary line, with another point on the end which is also infinitely small. We travel half the distance from one point to the other over and over again, yet we can never reach the other point, why? Because we have used bad definitions. If a point is infinitely small how can we have a point along the line which is halfway? You can’t. Because a discrete point can’t be infinitely small-we have created a contradiction before we even begin to solve the answer. So the contradiction can never be removed from the problem. All answers will necessitate a paradox.

Same thing with Cantor, the definition of a countable infinite set is a contradiction, there will always be another you didn’t count, just like in the set of real numbers. Infinitely large is the same as infinitely small, there is no beginning and no end.

@phoodoo:

Here are all all the natural numbers: N.

What natural number did I not “count” ??

No, there’s no contradiction. You’re just mixing up words. If you don’t like the definition, make up your own and call it “phoodoo countable” otherwise people will misunderstand you.

Countably infinite means a set lines up, one for one, with the counting numbers. It doesn’t mean you count them all which you could only do if the set was finite.

All,

Sometimes it’s helpful to take stock of where we are at. There seems to be three positions on infinity here.

1) Infinity does not really exist because it is not part of the phyiscal world and when explored deeply leads to paradoxes and is useless in mathematics.

2) Infinity is useful and necessary in mathematics but it’s a waste of time to think too deeply about it.

3) Infinity is useful and necessary in mathematics despite leading to paradoxes and not existing in the phyiscal world. This fact points to a real and profound existence beyond and above the material universe.

Does that fairly sum up the positions here?

Peace (on earth)

There have been a lot of nuts in mathematics and physics who suffered from some kind of neurological disorder such as bipolar disorder or autism. Cantor, Goedel, Einstein and many others were all nuts. This does not mean that they had nothing interesting or worthwhile to offer. But it is a sure bet that their mental illnesses are reflected in their works.

Only a total nut like Cantor would insist that two sets that can never be fully counted to establish their sizes can be compared to determine which is greater.

Only a total nut like Einstein would insist that a body can move in spacetime when anybody with a modicum of logic can see that it’s crap.

Only nuts insist that there is an infinite number of points on a line.

And I say this even though I, too, am a nut. I say it because it takes one to know one. 😀

PS. Don’t get me started on that hopeless lunatic, Goedel.

Jerad:

That is precisely the point. If you can’t count them all, the set does not exist.

That’s right, Joe! All those words – tree, morass, cardinal – have accepted meanings in set theory. But their meanings

withinset theory are completely different to their meaningsoutsideset theory.Which means that the comment of yours to which I was replying:

is utter bollocks.

Roy

One of the nice rewards of rejecting nutty sciences like infinite sets and the like is that, once you do, all the paradoxes disappear in one fell swoop. For example, once you reject the silly notion of continuity, all the paradoxes of Euclidean geometry (e.g., parallel lines that meet) simply disappear. Poof. It’s like having a huge load instantly knocked off your shoulders.

Mapou @ 336

If you mean parallel lines meet in projective plane, then yes they do, because all three types of conics (formed by the lines) are just a cone’s section in projective plane (For Eg- when you look at a painting).

JWTruthInLove@330,

Here are all the real numbers–N.

Which real number did I not count?

Its the exact same flaw.

Mapou:

How wide is a point, according to sane people?

JWTruthInLove,

What is half of your set, N? Its in infinity, right?

What half of that half, Also infinity, right?

Aren’t those two subsets also part of your set N? So there are an infinite number of infinities in your set N?

What is half of Cantors uncountable set? Its not countable.

I guess the countable set must be bigger. Heck, even one quarter of the countable set is bigger than the uncountable one.

Se what happens when you use nonsense ideas, you get nonsense. Mapou was exactly right.

Keiths,

Do you believe in infinitely small points?

Me_Think:

Only if you assume continuity which is hogwash.

keith s:

Once one realizes that distance is abstract, things like particle size, width, height, etc. do not exist. They are abstract concepts.

phoodoo:

Of course he does. He’s a crackpot. 😀

This thread is fantastic.

It could go on forever!!!

Some gems posted by our materialist friends.

1. One infinite number can be bigger than another.

2. An infinite number can be counted.

Hahahahahahahaha……….

No wonder you guys worship Darwin, you’re superstitious crackpots…….

Thanks for the laugh anyway….

logically_speaking:

Are you saying this thread is an infinite set? 🙂

Andre #346

Yes we are saying some infinite sets are ‘bigger’, have a larger cardinality than other infinite sets. When the idea was first proposed it was generally looked on with scorn by mathematicians but now it’s accepted and understood and used.

We are NOT saying an infinite number can be counted. Infinity is not a number and besides, if it were, you could always add one and get a larger one. Some infinite sets are said to be countably infinite because they are ‘the same size’ as the counting numbers (also referred to as the natural numbers or the positive integers).

But none of this has anything to do with materialism or evolutionary theory. Why would you lump them together? Because you disagree with them all?

I’ve known some very theological mathematicians who would take great offence at your bias. They might even say that as God is infinite it’s inspiring to study the infinite and to find that there too is great beauty and complexity.

But I guess you disagree. Your call. You’re missing some wondrous things, great beauty and great mystery. Fascinating.

Jerad

Spare me from your foolishness…. if its infinite it is infinite, there is no bigger or smaller…. OK maybe in your imagination. Adding one to infinite does not make it larger because you still can’t calculate it damn….

Infinite + 1 = ?

Infinite – 1 = ?

INFINITE…….

Please don’t patronize me, God is sitting this one out. Secondly it has everything to do with materialists and Darwinists because you lot act in such a way that does not fit the real universe. Prove me wrong, I dare you!

Andre #349

I am not trying to patronise you. KF knows what I am talking about so you can ask him. I’m just trying to explain.

If the universe was designed then surely when we see sets continuing on to infinity we should start down that path?

Infinity plus or minus one is infinity, surely. 🙂

Really though, there is a very elegant proof that there are more real numbers than rational numbers. Nothing to do with materialism. Not a thing. It’s part of mathematics, look it up.

5th,

There is another view, that we must accept that inherently ideational things can hold an abstract reality that constrains what is possible and even actual. For instance, there are no square circles as there is a logical contradiction of core attributes. So neither God nor us can bend a paper clip into a square circle. Such impossible beings cannot exist.

Logic — as abstract a thing as we get — constrains reality, and so holds reality in some form. (Which is a big clue. And no, I am not playing at platonism, I am highlighting that the evidence we have points to the fundamental power of ideas in reality, and as ideas seem inextricable from minds, to mind as a foundational aspect of reality. Mathematics, the Achilles’ heel of scientific-technical materialism.)

But, we can start with things such as how natural numbers are necessary beings, and that in a transfinite succession. More or less following a path trod by von Neumann (almost, as usual):

Strictly, ordinals so far, to get to cardinals, toss away the successor pattern, and for the nth in the sequence from 0, the cardinality of the number is effectively n – 1. That allows us to identify Aleph-null as holding cardinality of the set of natural numbers.

Note those pesky ellipses,

we are pointing to an in-principle, a supertask we cannot actually complete, but can contemplate logically— hint, hint on the contemplative, rationally envisioning mind. And, with a suitable set-builder procedure, we can appreciate that the natural numbers cannot not exist, appearing as a direct consequence of a successor process applied to the empty set and a cardinality assignment operation.Individual naturals are necessary and the set, which is patently transfinite, is also necessary, it cannot not exist in any possible world. Basic Arithmetic operations follow, per logic and things like 2 + 3 = 5 are necessarily true and in simple cases are self evident even to finite, fallible creatures such as we are.

We can take the number line as a useful construction, and look at the interval, [0, 1). We may define a proper fraction as a ratio of two natural numbers p:q, with q > a, interpolating in the interval. We may then transfer to the place value notation system, and might as well use base 10. Thus we see rationals as WHOLE + FRACTION, (partly) filling the gaps between successive numbers, 0, 1, 2, . . .

If we extend by allowing a series, and the usual notation, whole being W and fraction = B, W + B –> W.B, i.e.:

W + b1/10 + b2/100 + b3/1000 + . . .

With W = . . . w3 X 1,000 + w2 x 100 + w1 x 1, any number on the line can be expressed

. . . w3w2w1.b1b2b3 . . .

where, the ellipses indicate infinite series. Countably infinite, i.e. we apply as many digits as are in a set of cardinality Aleph-null. (In many cases we will have a huge array of leading and trailing zeroes. Which, from grade school on we usually ignore.)

It can be argued that for any given W.B1 and W.B2, we may interpolate another number, for convenience, the average of the two, or some intermediate at any rate. Thus, again in principle, we fill in the spaces, so that gaps between the rational numbers are covered. That’s where irrationals and transcendentals lurk, and once we identify any given one, we can arguably show that each is a root of at least a countably infinite set of others, by applying multiples and fractions etc. pi, 2*pi, 3*pi etc and pi/2, pi/3 etc.

So, we have a case of our finitude and pointing to what lies beyond finitude.

The numbers we can actually directly handle and compute out to actual expression form a fine dust on the real number line, but the line is continuous per the concept of interpolation. And, extends without limit.All of this is abstract, and in fact we have introduced continuous variables, i.e. W.B is a continuous real variable quantity. Toss in additive inverses headed the opposite direction and negatives are there too. Pay off a debt – d by depositing d, and you owe 0. That’s where these first were recognised in our civilisation.

This can be assigned, x if we please. To get y, I suggest the complex number approach where i*i*x = – x and plausibly i*x –> y, an orthogonal axis. This allows a 2-d space, with angles defined algebraically by Z = x1 + i*x2, with r^2 = x^2 + y^2 and tan h = x2/x1, etc. We can extend to the ijk system for 3-d space and by viewing t as time, get to trajectories x(t), y(t) etc.

Physics can be added, and so forth.

But the point is, we have an abstract set of concepts with logical constraints implicit that apply to physical reality.

And, this brings in the in-principles of those ellipses. Continuous, abstract variables in a mathematical world that models the physical one we experience, with a gateway from the logic to constrain it. And with the transfinite there, at both ends, the infinitesimal and the ultra-large.I hardly need to say, such has been highly successful in sci-tech and more mundane pursuits alike.

But at the root of it is a ghost, the suggestion that mind sits at the root of reality, that the ideal forms we have discussed insofar as the finite and fallible can (courtesy ellipses) are inherently mental and eternal. Where, had there ever been utter non-being, that would forever obtain as non-being has no causal power.

Eternal mind, eternally holding and contemplating such abstract realities.

The ghost of Augustine is laughing.

And, while I can appreciate why many are concerned on issues of crack-pottery, last I checked, all of us are cracked pots, and there is One who specialises in that:

The glory from beyond shines out through the cracks . . .

(More ellipses . . . )

KF

@phoodoo:

What flaw?

You’ve mentioned a contradiction. What’s the contradiction?

Mapou @ 342

If you don’t assume continuity, then a line is no longer a line -it is a line segment.

Andre, I understand the tendency to suspect authority (especially ideologised authority) in our day. And, we have been burned by the hot stove a few times. But our response should not be as Mark Twain’s cat, which would never sit on a stove again thereafter — hot or cold. There is much to object to, and the issues of the strange difficulties of the transfinite lurk, but that should not lead us to hyper-skeptical dismissiveness. At minimum, let us respect serious, empirically highly reliable work, even if we have reservations on points that look doubtful. Yes, the abstract continuum is riddled with the transfinite, but so are ordinary counting numbers and rationals; things that we can only contemplate with ellipses. Yes, there are horrors there, which led the ancients to erect forbidden zones. But, with reasonable processes — and yes, Cantor paid an awful personal price — we can get a few glimpses that allow us to have a higher confidence in our work than otherwise. BTW, that was also how I came to swallow the issues of quantum and relativity, green eggs and ham stuff that were pretty hard to deal with. KF

MT, any finite line segment is a continuum, with transfinite cardinality. We contemplate here, the shocking issues that lurk in so simple an exercise as addressing a line between points. And, back in the day, the debate on how many angels can dance on the head or point of a pin was about location vs extension, i.e. the infinitesimal contemplated by defining a point as pure location sans extension. The world is stranger than we imagine, perhaps stranger than we CAN imagine, and I don’t remember who I first saw with that one. KF

KF #354

Extremely well put!!! Thank you.

Very we’ll put indeed. I may ‘borrow’ that paragraph.

~Jerad, in the cold, no warm water or beaches for moi. Cool shiverings mon.

Ph,

pardon but kindly look at 351, on the uncounted, uncountable numbers, the irrationals and transcendentals, with pi and e as familiar cases, a lot of logarithms and trig vales being suspected cases, and more. They probably outnumber the rationals and integers.

Some argue as 2^Aleph Null to Aleph null or thereabouts. (There is a debate as to the relationship between c the continuum number and the successive power set scales on aleph null.)

Wiki:

Here, there be monsters . . .

KF

Jerad, feel free to cite. And the solution to a freezing New Year’s is as close to hand these days as your friendly travel agent, but of course even with Brent just now at $ 60.24 /bbl, that’s not cheap. Funny how $60/bbl looks cheaper going down than it did going up, but then the inflationary impact of the past several years should not be underestimated. I only hope we have an era of fairly cheap energy ahead, I think the Saudis only have so much influence and face the issue that shale and fracking stare them in the face. Oh for LIFTR-Th and real fusion. KF

KF #358

Too little money and family responsibilities preclude travelling I’m afraid but I would very much like to visit the Caribbean some day. I’ve seen pictures, clear, warm water to die for. Sigh.

Jerad, I understand. Many’s the day I have had to park next to just such a beach, then glance wistfully at the waves, and turn away to deal with the issues of the day for a client or two. KF

JWTruthInLove,

You said:

“Here are all the natural numbers: N.

What natural number did I not “count” ??”

If you can say this, and think it makes sense, why can’t one also say,

“Here are all the REAL numbers: N.

What real number did I not count? ”

Is it true for both cases or true for neither?

Jerad,

Are there or are there not an infinite number of infinities in the set of all natural numbers?

Phodoo,

The natural numbers (aasuming you mean the positive integers) are countably infinite, have cardinality aleph-0, the “smallest” infinity.

So no, the natural numbers do not have an infinity of infinities.

The real numbers are another matter and not the same as the natural numbers.

Jerad,

Then how large is the subset, in the set of natural numbers that contains half of them?

Mapou:Calling something continuous does not make it so.The continuum is a mathematical model. Between any two distinct points on a line, there is a point in between, and there are no gaps (least upper bound property).

phoodoo:A set of infinite numbers is already uncountable!In mathematics, countable is defined as having a one-to-one correspondence to the natural numbers or to a subset of the natural numbers.

Mapou:The point is that, regardless of how far you zoom in, it is still discrete numbers.The Euclidean plane and Newtonian manifold are continuous by mathematical definition.

phoodoo:Once you have told me all of the elements of your “countable infinite set” I can then give you another one you didn’t count.Consider the set of natural numbers.

fifthmonarchyman:3) Infinity is useful and necessary in mathematics despite leading to paradoxes and not existing in the phyiscal world. This fact points to a real and profound existence beyond and above the material universe.What paradoxes?

Mapou:If you can’t count them all, the set does not exist.In set theory, there is the set of natural numbers.

Mapou:all the paradoxes of Euclidean geometry (e.g., parallel lines that meet) simply disappearParallel lines never meet in Euclidean geometry.

phoodoo:So there are an infinite number of infinities in your set N?That is correct. If you divide N into subsets, at least one of them must be countably infinite. Indeed, there are uncountably many countable subsets of N.

Mapou:Only if you assume continuityAssume there is an additive identity.

Andre:if its infinite it is infinite, there is no bigger or smallerCantor proved that the real numbers are a larger cardinality than the counting numbers.

Me_Think:If you don’t assume continuity, then a line is no longer a line -it is a line segment.Actually, there is a one-to-one mapping between the line and the line segment, or the line and Euclidean space, or if you prefer, the lotus blossom and the cosmos.

phoodoo:Are there or are there not an infinite number of infinities in the set of all natural numbers?If you divide N into subsets, at least one of them must be countably infinite. Indeed, there are uncountably many subsets of N.

Phoodoo,

It’s also countably infinite oe aleph-0.

LoL! @ Roy:

Why is that, Roy? When words to not have a specific meaning in specific scenarios it means their meaning remains the same throughout.

Two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set.

For example let A = {1,2,3,4,…}

Let B= {2,4,6,8,…}

We can remove every element of set B from set A and still have an infinite set left. So how can the two sets have the same number of elements?jerad,

And how large is the subset that contains 1/4 of all natural numbers? Infinite

And how large is the subset of the subset which contains half of all natural numbers? Infinite

And how large is the subset, 1/1000 of the subset that contains half of all the natural numbers? Infinite.

So we have just confirmed that the set of natural numbers contains an infinite number of infinities.

Why did you just deny this?

Zacheriel,

You sound like a typical science skeptic; you believe anything someone with a title tells you.

Cantor proved nothing.

phoodoo:And how large is the subset that contains 1/4 of all natural numbers? InfiniteCountably infinite.

phoodoo:And how large is the subset of the subset which contains half of all natural numbers? InfiniteCountably infinite.

phoodoo:And how large is the subset, 1/1000 of the subset that contains half of all the natural numbers? Infinite.Countably infinite.

phoodoo:So we have just confirmed that the set of natural numbers contains an infinite number of infinities.Not just infinite, but uncountably infinite. Per Cantor’s theorem, the cardinality of the power set of A (the set of all subsets of A) is greater than the cardinality of A.

phoodoo:you believe anything someone with a title tells you.Not at all. Cantor published mathematical proofs.

Joe,

That’s the way countably infinite sets work. The positive integers (countably infinite) take away the evens (also countably infinite) leaves you the odds (also countably infinite).

It takes a while to get used to it. But, like relativity and quantum mechanics, it starts to make sense eventually.

Jerad:

Then the two sets do NOT have the same number of elements. Thank you.

Phoodoo,

I thought you meant that in a different way. And many of ‘infinities’ you are thinking of overlap. I suppose what you say is right although i wouldn’t say it the same way.

But yes there are a lot of countably infinite subsets of countably infinite sets. Again, the criteria for countably infinite is: can it be matched 1-to-1 with the natural numbers.

Jerad:

That’s the trick, isn’t it?

Two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set.And that also means there isn’t any real 1-to-1 correspondence.KF

Indeed, you are not incorrect, but on a serious note, and I stand by this, materialists, do not act in a way that fits the real universe.

Darwinism, multiverse, and this pile of junk we’ve been dealing with in this thread. Sure the mind can imagine all kinds of things, but that does not make it so.

Joe,

You can put the positive integers and the positive even inters into a 1-to-1 correspondence so, even though one set contains the other and has more elements they both have the same cardinality. All of the following are countably infinite:

The integers, the positive integers, the multiples of 2, the multiples of 3, the multiples of any counting number, the perfect squares, the perfect cubes, the primes (not sure about the prime pairs yet), the rational numbers (all the possible ratios of integers). All if these sets are the same ‘size’, the same cardinality. It is weird to think about at first. But all of the above sets can be put into a numbered list which will leave none of the set out. And the numbering puts the set into a 1-to-1 correspondence with the counting numbers.

I’m sorry if you don’t like it or think it’s rubbish. But it’s established, accepted and it works.

Jerad:

Not really and I have explained why. That you have to ignore that explanation tells us that you cannot deal with it.

So cardinality doesn’t refer to the number of elements in a set? I was always taught that the cardinality of a set is the measure of how many elements it contains.

It isn’t used for anything, Jerad. So how can you say that it works? Perhaps you mean it works for lazy people who don’t want to think about it and want and easy way out of the obvious contradictions.

And by “established” Jerad means “accepted by those who are too afraid to stand up and be counted”

Jerad:

That is your opinion and it does not amount to proof in a mathematical sense.

It is easily refuted, so yes “weird” isn’t the right word.

Good luck with a numbered list for a set of infinite numbers. Let me know when you are finished and we will check your work

Here’s another way to think about it:

Can you write down a pattern for your infinite set so you’re sure that following the patter nothing will be left out?

Here’s a sequence: 2, 5, 8, 11, 14 . . .

The pattern is: add three to get the next term.

Here’s my 1-to-1 correspondence with the counting numbers . . .

The counting number n is matched with 3n – 1.

n = 1 is matched with 2

n = 2 is matched with 5

And so on.

You can give me a counting number n and I can tell you what term of the sequence it’s matched with (n = 23, for example is matched with 68). Or you could give me a term in my sequence and I can tell you what counting number gets matched with it (110 is in my sequence and it’s matched with the counting number 37).

So now essentially I’ve got the elements of the two sets paired and no element of either set is excluded. The sets must be the same size or some element of one set would be without a partner.

Joe,

No matter how many aspersions you choose to cast you will find what I’m telling you in many textbooks and web pages. Please go have a look. It is not a lazy way out and there are millions of people who are standing up and being counted as supporters, KF among them.

Jerad:

I never claimed otherwise. I am claiming that doesn’t make it right.

And thank you for continuing to avoid my arguments. That alone is very telling.

And that tells me the difference in size between the two sets. Not my fault that you are stuck in a box.

The 2n required for positive even integers and positive integers also tells me the difference in size between the two sets.

Joe #382

I am trying hard to answer your questions even though they are stated in non-standard forms.

Except it’s just formula relating one element to another. What does that tell you about their relative sizes? Assuming you’ll say one set is three times the size of another how can you prove that? My formula shows that every element of each set is married to an element of the other set. No element is unmarried. How could that be if one set was larger than the other?

Find me an element of one set that is not married to an element of the other set. That has to be true if one set is bigger than the other set

This is where your approach fails. You cannot pick an element in one of the sets that is not married to an element in the other set. There are no bachelors and no polygamous elements eiher. The sets are matched up, one for one. How can that happen if they aren’t the same size??

Just find me an element of one set that isn’t married, that’s how you falsify my correspondence. That’s all you have to do.

Jerad:

Heh.

Heh. I say your formula shows the difference in sizes between the two sets

The formula. And it’s 3 times minus 1 in the scenario you are referring to.

And yet I have. Here it is, AGAIN:

Two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set. And that also means there isn’t any real 1-to-1 correspondence.Copernicus, 16th century- “The earth revolves around the Sun”

Jerad- “No, look at all the textbooks and references that say the sun revolves around the earth”

Joe,

Set 1= {1, 2, 3, 4 . . .} Set 2 = {2, 5, 8, 11 . . . } I marry all the elements of the first set with elements of the second using the formula

n from the first set marries 3n – 1 from the second set

If one set is bigger than the other then you should be able to find an element of the bigger set that is not married to an element from the smaller set.

Can you do that? If you can’t then how can the sets be different sizes?

Your assertion is meaningless if you can’t find an unmatched element. You say there must be one at least. Name one.

I think I’ve found a 1-to-1 matching that leaves no element unmatched. Prove me wrong.. Find an unmatched element and prove one set is bigger than the other.. Just one element will do.

Joe #385

Prove me wrong Joe. Find an unmatched element from one of the sets using my matching rule. If you can then my 1-to1 correspondence is wrong and one set must be bigger than the other.

All you have to do is find one unmatched element.

We are arguing with demon possessed lunatics on the one hand and dishonest jackasses on the other. I hate repeating myself but here goes.

Joe:

Joe,

What is the “difference in size” between these two sets?

A = {0,1,2,3,4,5,6…}

B = {0,1,4,9,16,25,36…}

Once again, one cannot compare two sets to determine which one is bigger unless one determine their sizes. Since nobody can determine the size of an infinite set, saying that infinite set A is bigger or smaller than infinite set B is pure hogwash from a deranged mind. Somebody should go over to Cantor’s grave and defecate on it. Then his followers should be tarred and feathered and paraded in the streets. 😀

Ampoules #388

Read my post 386 and see if you can find an element of one set that is not married to an element of the other set by my rule.

This is the kind of thing Cantor worked on. He set up similar 1-to-1 correspondences and he couldn’t find an unmatched element. So he figured the sets must be the same size. Infinite but the same size.

Mapou:

keiths:

Mapou:

Speaking abstractly, how wide is a point, according to sane people? How many points are there on a line segment that is exactly one inch long?

Jihadist @391,

Anybody who claims that an infinite set has a size is suffering from some deep mental disability. And we all know Cantor was a fruitcake and so are his moronic followers. 😀

keith s:

Why speak abstractly and why ask me? I am not one of the idiots who claim that there is an infinite number of points on a line. I don’t believe points exist. I believe there are only particles, their properties and their interactions. Everything else is either abstract or voodoo nonsense.

Now, if you want to know how many particles can fit on a 1 inch segment (the abstract distance between two particles), one can easily calculate it by dividing 1 inch by the Planck length. It’s a very huge number. Here I’m assuming that the Planck length is the fundamental discrete unit of distance. There is a good chance that it is not because it was calculated through dimensional analysis and some people are not comfortable with that.

Mapou (sorry for the predictive text typo above) #393

Cantor looked at set matches like what I’ve listed in #386. He realised he couldn’t find mismatched elements in some pairings. He concluded that there were a whole class of sets that were infinitely large but the same size.

And then he found one where he could get a mismatch. Where the pairing could not be made to work. What could he do? He’d found an infinite set that WAS bigger than other infinite sets. There were different infinities.

Mapou #394

You are the real materialist!!! 🙂

I’m not really sure I should bother asking but, for anyone who agrees with this:

Which point on a number line is pi? What are the points immediately left and right of pi?

And lots heat and very little light has been created here over the meaning of the size of set. Folks that want to

get to grips with the math might want to think about what size actually means for a set. There are (at least) two different concepts that describe a set’s size.

The cardinality of a set is the number of elements, and as others have pointed, it’s possible to show that the set of all integers has the same cardinality as the set of all odd integers.

There is also the density of set, when it considered a subset of some other set. So, when considered as a subset of the set of all integers the set off all odd integers has density of one half. So two sets can have the same cardinality when they considrered as sets in their own right, while having a different density when they are considered as subsets of another set.

These ideas might not be very intuitive, but they are important and powerful ideas in math, and it’s worth spending some time trying to understand them.

Jerad:

Jerad, if I am questioning the claim that you are marrying all of the elements using that formula then you cannot use that formula as an argument that you are marrying them.

You have no idea how to engage in a debate that challenges the orthodoxy. If the orthodoxy is being challenged then it cannot be used as evidence in support of itself.

Also this:

Two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set. And that also means there isn’t any real 1-to-1 correspondence, is a fact and not an assertion.What is the “marrying formula”? The “marrying formula” = the difference in size between an infinite set and all of it’s infinite proper subsets.

Jerad:

I don’t care how the fruitcake redefined terms such as size, bigger and smaller. I don’t give a rat’s asteroid. All I am interested in is the demise of the braindead idea that nature is continuous.

wd400:

And it’s possible to show that the set of all integers has more elements than the set of all odd integers. Oops.

What, with a subset the elements have more space between the commas? 🙂

What import and power would that be, exactly? I have been asking for a long time and no one ever answers except to say it is important and I am ignorant. Not very convincing especially given the fact that I have refuted their claims and all they can do is repeat them without addressing the refutations.

Joe #399

My ‘rule’ is my claim. So, yes, I can use it.. Prove my rule wrong!!!

I say: take an element n from {1, 2, 3, 4 . . . }. Marry/match n to 3n – 1 in {2, 5, 8, 11 . . . }

Can you find an element in set 1 that is not matched to an element in set 2 by my rule? If you can then I will agree that set 1 is bigger. That’s it.

Set 1 has all the elements of set 2 and more. So set 1 should be bigger than set 2. So my rule matching set 1 to set 2 shouldn’t work. So you should be able to find an element in set 1 that does NOT get matched with an element in set 2..

Can you do it?

All Cantor the fruitcake showed is that if one counts or expands abstract series, some series will grow faster than others. Big effing deal. Who cares? He had nothing new to say about

infinitythat we did not know.Mapou:All I am interested in is the demise of the braindead idea that nature is continuous.Nature may not be continuous, though string theory is still tentative. However, we were discussing the mathematics of the continuum.

Jerad:

I have, using the mathematics of sets. That is by subtracting all the elements of one set from the other we, together, have demonstrated there are elements left in one set but not the other. That means there are elements left unmarried.

And using the same mathematics of sets I have shown that your formula actually shows the difference in size between two sets with infinite elements. I know that upsets you but too bad.

Joe #402

Let’s just work with the positive integers.

I = {1, 2, 3, 4. . . }

O = {1, 3, 5, 7 . . . }

n is an element in I. n gets matched with 2n – 1 in O.

I claim that this matching shows that I and O are the same size because no element of I or O are unmatched. Give me an element in I or O and I can tell you the unique element of the other set it is matched to. It’s a 1-to-1 correspondence.

If you’re right there should be at least one unmatched element. Find one and I’ll concede.

Jerad:

LoL! Of course you can use it. But you cannot use the rule as evidentiary support for the rule. And that is all you have been doing

Jerad, Obviously you have issues that severely limit you, intellectually.

There are an infinite number and using the mathematics of sets we have found them. That is by subtracting all the elements of one set from the other we, together, have demonstrated there are infinite elements left in one set but none the other. That means there are infinite elements left unmarried.And using the same mathematics of sets I have shown that your formula actually shows the difference in size between two sets with infinite elements. I know that upsets you but too bad.Joe #406

I’m using a different matching than you. But, if you’re right you should be able to find an element that isn’t matched. If you subtract elements from the second set by my scheme, by taking them out as pairs you don’t get anything left over.

Explain how that happens without claiming that one scheme is ‘better’ .

Show where my scheme fails.

I take n out of the first set and 3n – 1 out of the second set (if we’re

Talking my first example) as a pair. What gets left behind? Find an element that doesn’t get taken out.

Jerad:

Yes, you use a false matching.

There are an infinite number and using the mathematics of sets we have found them. That is by subtracting all the elements of one set from the other we, together, have demonstrated there are infinite elements left in one set but none the other. That means there are infinite elements left unmarried.And using the same mathematics of sets I have shown that your formula actually shows the difference in size between two sets with infinite elements. I know that upsets you but too bad.It is possible to show, and has been shown, that the set of positive integers has more elements that the set of positive even integers. If Jerad et al., were correct then that should not be possible.

Set 1: The grains of sand in the Sahara desert

Set 2: The M&Ms in an unopened packet.

Most people would have no difficulty whatsoever deciding which set is bigger without determining their sizes – but mapou thinks it’s not possible.

For a less extreme example, take a box of bolts and a box of washers. You can easily find out which set is larger by threading washers onto bolts until one of the boxes is empty. Mapou thinks this is impossible too.

Not that I expect mapou to admit he is wrong.

Roy

We can determine the size of the set of an unopened pack of M&Ms.

Roy:

Haysoos Martinez! What is wrong with the mental midgets of the materialist/Darwinist camp? I’m truly running out patience with that clueless bunch. This is like saying that just because I don’t know the exact number of stars in the universe, I cannot conclude that there are more stars than people. There is such a thing as a size estimate, you know. There are logical, well known ways to estimate the size of various finite sets. The problem is that there is no logical ways to estimate the size of infinity. It is ludicrous on the face of it.

I am fast coming to the conclusion that infinity mongers, Darwinists and materialists are all mentally ill. Hell, we all know that Darwin was a fruitcake, just like Cantor, Godel and Einstein. I am arguing with an insane asylum. Lord have mercy!

I’m going to invoke Proverbs 26:4

There is nothing more infuriating than an anal retentive fruitcake.

Can someone explain to me how to determine the size of a set?

Thanks.

Mapou

I can’t agree more crackpots come to mind…… We can calculate infinity and there are bigger and smaller infinities…..

Hahahaha, and these are the people with logic and reason on their side…….

Hahahahaha!!!!!

Mung:

Textbooks say by counting the number of members the set has. However, some people measure the distance between the {} and others just make bald declarations.

I am a card-carrying Yin-Yang dualist. I believe that opposites are ONE. There can be no such thing as left without right, up without down, open without closed, no without yes, first without last or beginning without end. If one exists, its opposite also exists. To believe in infinity is to deny the logic of complementarity. The infinite has a beginning but no end. Infinity mongers are a scourge on civilization. They retard the progress of science by centuries if not millenia. They should all be placed in a mental institution for their own protection and the protection of society, and away from kids and animals. 🙂

Yes. It is. But that is a consequence of what

yousaid. If it is ridiculous, you only have yourself to blame.Yes, I know that. It makes no difference to what you said.

Yes, estimating the size of infinity is ludicrous. So ludicrous that either every mathematician is a blithering idiot, or you are completely missing the point.

Based on your dismissal of the entirety of infinite set theory as hogwash, and you description of Cantor as a fruitcake, I expect you think every mathematician is a blithering idiot.

But in fact everyone else is laughing at you and Joe.

Roy

Count the number of elements. If you can’t do that, find a rule that allows a one to one mapping to a set of known size. (If you knew how many people where in a room you could establish if the number of chairs was equal to the number of people just by asking them to sit down).

One the examples above is the sets {0,1,2,3,4 …} and {0, 1, 4, 9, 16 …}. It shouldn’t take too long to see there is a function that maps an element from the first set to one and only element in the second set (the square of every element in the first set appears once in the second), if we can map elements between these sets in this way we know they are of the same size (just like the one-to-one realtionshp between bums and seats let us establish to number of seats in the example above).

In fact, the first set is a special case, the set of all natural numbers (+ve integers), and the cardinality of that set is called aleph_null (indeed the square numbers are the first example of a set with this cardinality on the wiki page for this concept).

Roy @422,

Roy, you anal retentive slut! 😀

This was my rendition of an SNL classic.

Mapou,

I don’t know much about physics, but I do believe that nature is most likely not continuous, FWIW.

But remember that not all mathematics exists to serve physics or the sciences. What is wrong with working with mathematical concepts that don’t correspond to physical reality if they are interesting and/or useful?

For example, what if someone asked you to calculate the integral of sqrt(1 – x^2) between x = 0 and x = 1.

Is it ok to give pi/4 as the answer, or would you insist that only a rational approximation is acceptable?

Hey, Joe!

Are these two sets the same size?

Set 1: {1,2,3,4,5,…}

Set 2: {-2,-4,-6,-8,-10,…}

How about these?

Set 3: {1,2,3,4,5,…}

Set 4: {1,(1),((1)),(((1))),((((1)))),…}

Or these?

Set 5: {1,2,3,4,5,6,7,…}

Set 6: {1+2,2+2,3+2,4+2,5+2,…}

Set 7: {3,4,5,6,7,…}

And finally:

Set 8: {1,2,3,4,5,…}

Set 9: {(1,1), (2,2), (3,3), (4,4), (5,5),…}

Set 10: {(1+1), (2+2), (3+3), (4+4), (5+5),…}

Don’t disappoint.

Roy

keiths:

Joe:

Seriously? You can’t find one?

Okay. So answer my question:

Read it again, Joe:

A lesson in cardinality for Joe GI really have to agree with Mapou, talk of preposterous infinity conclusions shows the extent to which many materialists are willing to forgo any sense of thought, in the name of believing in a scientific paradigm.

Indeed Cantor was nuts. Perhaps it is one reason why he was unable to see that numbers (which ever kind you want to call them) are creations in man’s mind. As such they don’t have size. They don’t have cardinality. They aren’t limited or unlimited, except in the sense of what any one person can imagine.

Saying that I can put all of my imaginations about one idea in one set, and that imagination is bigger than another imagination that I decide to call something else in another set is smaller, but unlimited, is surely crazy talk.

Why is the set of all real numbers bigger than the set of all M&M’s I can imagine? Its not. Because since the set doesn’t actually exist, except in my mind, I can make the set any size I want. So my M&M set is bigger. And I can even do a mathematical proof just as accurate as Cantors. Just put all of you real numbers into a graph anyway you choose. Now, I reserve the right to create in my mind one kind of M&M that I will call uncountable, and I draw it on the page, with a symbol, outside your graph. Since you have no one to one correspondence to my M&M, my set is bigger.

That is all Cantor has done. And crazy people (mostly materialists it must be said) take this as fact, and are not willing to use their minds for even just a few seconds to see how dumb it is. Its a collective willingness to be told what to think.

One imaginary set is not bigger than another imaginary set, any more than the set of “funny” is bigger than the set of “enlightened”.

daveS:

I have no problem when fruitcake mathematicians play in their sandboxes. I do have a problem when they want to impose their sandbox philosophy on science. When that happens, we get sandbox monstrosities such as time travel, infinite parallel universes, black holes, continuum physics, the relativity of motion and position and other similar hogwash. I have no problem with using Pi/4 but I do have a problem with BS like division by 0 and sqrt(-1). I do have a problem when some lunatic insists, in or out of the sandbox, that there is such a thing as an infinitely smooth circle or an infinite set. Infinite anything is crap because it violates the principle of complementarity in that it posits the existence of a beginning without an end. This is absurd. Now, if the fruitcakes had properly labeled their sets with names like ‘progressive sequences’ or ‘expanding series’, I probably would have no objection.

Take the probabilistic nature of particle decay for example. Why is it that particles do not have fixed decay intervals? Physicists have no clue as to why that is. And you know why? It’s because they’ve allowed a bunch of fruitcake mathematicians to step out of their sandboxes and play with grownups. That’s why.

Phoodoo:

They are an elitist group of jackasses and fruitcakes who have managed to convince themselves that science belongs to them and that nobody else but them have the right to conduct science.

Mung: Can someone explain to me how to determine the size of a set?

Joe: Textbooks say by counting the number of members the set has.

wd400: Count the number of elements. If you can’t do that, find a rule that allows a one to one mapping to a set of known size.

ok, that’s sort of what I figured. Thanks.

So does it make any sense to speak of the size of an infinite set? Is the size of an infinite set undefined or just nonsense?

Mapou @ 430

Do you realize that Imaginary number [Sqrt(-1)] is fundamental in calculating the wave functions of Quantum particles? Without imaginary number, there is no QM. Division by 0 is used in many complex analysis (‘Complex analysis’ is a branch of mathematics)

Mapou,

Thanks for the reply. For the record, I also have a problem with division by 0!

I’m not sure I understand how you arrived at your position on pi/4, however. How do you even define pi in your system?

It seems that Keiths has balked (chickened) out of answering if he believes in an infinitely small point. Do any other materialists wish to take a stab at this? Does such a thing exist?

Mung, there are indeed good reasons to talk about the size of an infinite set. In particular because some infinite sets are larger than others. There are more reals than natural numbers, for instance.

Mapou,

I am equally offended by how a small group of not very clever thinkers has managed to pull the wool over the eyes of so many others in the general public, to the point that virtually all of the knowledge you can receive online or in print, is really just these weird scientific skeptic agenda, that is masqueraded as some kind of truth.

All these things on youtube, and on so called science shows, or podcasts, or science forums, to the level of National Geographic, or the BBC or major newspapers, they talk about things like Einstein, or evolution, in a way that suggests that they have a clue about the reality of this information, because it has become part of the cultural fabric of society. When Penn Jillete goes on a rant about evolution, invoking the “great minds” of people like Richard Dawkins, he speaks as if he knows he is right about the subject, because all he is ever exposed to is sources he trusts, which tell him its true. If people like Bill Nye, and the BBC or the New York times write articles about how obvious evolution is, well why wouldn’t he just go along and believe this. So then you have people on the Tonight show, or Saturday Night live, or Huffpost talking about all the dumb people who don’t believe evolution, because well, the celebrities even know its true, they read it!

And where did the writers for the BBC and the New York times get their information? Well, its just out there, everywhere. Or from Jerry Coyne, or Neil Shubin, or Danniel Dennet. Did they understand the voracity of the claims, of course not, but so what. Its everywhere in the public, so go ahead, it must be true-everyone knows it.

Are Cantors sets real, of course, its taught in universities, it has to be true. Don’t you know the only people who doubt it are cranks? Just look it up on the internet, there are all kinds of people online telling you they are cranks, why think?

We are a nation of mass media fools, skeptics who absolutely refuse to be skeptical of anything they are told.

wd400,

see above post. oh brother…

If there were more real numbers than numbers you can count with, you could never know it anyway, because you wouldn’t have the capacity (the counting numbers!) to be able to count to check.

“Oh no, we have run out of the ability to say, oh, there is another one, and another one…”

Infinite Sets and Infinite Sizes

This should probably be taken with a grain of salt. After all, it comes out a philosophy department.

Earth to keith s- anyone who claims that unguided evolution predicts an objective nested hierarchy is mathematically deficient and has no business trying to correct me in anything math related. And anyone who bitches at me about finite vs infinite and then uses finite sets to make a case for the infinite, is a total whack-o.

I made my case, keith s. You can either deal with it or continue to ignore it. Onlookers aren’t the fools that you think or hope they are.

What do you think is wrong with Cantor’s diagonal proof?

And keith s- please do TRY to keep up. Jerad introduced the “marrying formula” and my claim is that formula provides the relative cardinality difference between an infinite set and all of its proper subsets.

If you want to know the difference in size between your two sets then figure out the “marrying formula”. I have tried to help you with nested hierarchies, which are related to set theory, and you refused my help and continue to bastardize the concept. There is no way I will listen to you wrt set theory. And that goes especially when you blatantly ignore what I say in defense of my claims.

What’s wrong with you?

Hi Roy- Guess what? You can use my methodology that you quoted and figure it out for yourself!!!11!11!!!1!!! When comparing negatives to positives just use absolutes to make it easier on yourself.

Joe, you are at sea on this I’m afraid. The cardinality of any infinite subset of the natural numbers is equal to the cardinality of the natural numbers (aleph_null).

wd400- Deal with my argument or are you will prove that you are afraid.

phoodoo @ 437

You can write a paper and publish to establish your self as the most learned guy. No one is stopping you. Do you have a counter proof for any concepts you want to negate? Publish it. Afraid of peer review? Self Publish. Let the

learned majoritybe the judge.Me_Think:

I don’t care. It’s obviously a chicken shit kludge since Sqrt(-1) is BS on the face of it. What QM physicists should explain is what causes this so called wave function. They have no clue.

I’m equally unimpressed. I have always been of the opinion that any phenomenon that requires a lot of weird math to solve is one that physicists are completely clueless about. I would say that 90% of physicists have no idea that nothing can move in spacetime. Hell, when told about it, they immediately deny that it’s true.

The ignorance of physicists is deep. I remember the faster than light neutrino fiasco in 2011 involving a whole slew of highly paid physicists at CERN and elsewhere. It was embarrassing to say the least.

daveS:

There is nothing wrong with Pi. One should calculate Pi to whatever precision is required by the problem one wants to solve. Pi is the perfect example of the stupidity of continuous structures. Nobody can fully calculate Pi simply because continuity is nonsense. In nature, there is a limit to how smooth a circle (or any curvature) can be and this limit is enforced by the Planck length, i.e., the smallest fundamental distance between two particles.

Set 1 = {1, 2, 3, 4 . . . }

Set 2 = {3, 5, 8, 11 . . . }

n in set 1 gets matched with 3n – 1 in set 2. It’s just matching them up in order of appearance.

My matching links 1 in set 1 with 2 in set 2. Take those two out.

Then 2 in set 1 gets matched with 5 in set 2. Takes those two out.

Continue with 3 and 8, then 4 and 11. So far no element has been unmatched.

Continue on . . . I have shown a way to match each element of set 1 with a unique element of set 2. No element is unmatched. Give me an element in either set and I can tell what element in the other set it is matched with. You can’t just say: you can’t use that without quoting a set theory rule which says it’s not allowed.

If the sets are different sizes then there should be at least one unmatched element somewhere. If you want to disprove my scheme all you have to do is find an unmatched element.

If you can’t find an unmatched element then the sets must be the same size.

If you think there is any mathematical reason my scheme is wrong then please give me a mathematical reference clearly stating so.

Joe #442

All my scheme does is match elements of one set with the elements of another set.

You look at the 3n part and think that says some thing about the cardinality. But you’ve got no mathematical references to back up your claims or your objections. What does relative cardinality difference mean anyway? Give a mathematical definition of the term please. And don’t just say the words mean what they always mean, you know that many disciplines use terms differently and in a specific way.

With my scheme no element of either set goes unmatched with an element of the other set. If there were an unmatched element then then sets would be different sizes. Find an unmatched element and you win.

I’m assuming you’re having a hard time finding an unmatched element or you would have presented it by now.

Show me where my scheme fails.. Don’t just say it does this or that, don’t just say it’s contrived without backing up your use of that term with references. You always ask me for references but you’re very reluctant to provide you own in this case.

Again, find an unmatched element, that’s all you have to do..

jerad,

The exact same logic can be used to compare the set of natural numbers with the set of real numbers, how can you not see that?

What is the first number in the set of real numbers? I will choose to match the number 1 with that. What is your second number? Since you can’t even say what the second number is, the problem isn’t in the number of elements in any one set, the problem is in you defining what elements are in your set, and what elements aren’t. As soon as you can tell me two of the elements, or three of the elements, then I can tell you what the matching number is from my set of natural elements-its as simple as that.

There is nothing that says I have to match up a one in my set with a one in your set. I can just as well match the number 1 in my set, with the number for Pi in your set. Ok, next….

You would have to be a crazy person (Cantor) to not see how easy this idea is.

Andre, 376:

You have a point, e.g. at one level we can verbalise, “square circle,” and then we may analyse further and see, not a possible being. That comes about because the core characteristics stand in mutual contradiction. Not even God could make a square circle.

But in turn, that brings to bear how inherently mental conceptions and contemplations can have powerful impact in the real world. Square circles etc are forbidden beings, but also — as I argued in 351 — natural numbers are necessary beings, automatically present in any possible world:

From this, I continued to show how we may construct the real number line by recognising (using decimals for convenience, many bases are possible) that W.B, whole + fraction, extends this powerfully:

So, we may then go for negatives by the concept of additive inverses: (- r) + (+ r) –> 0, which is the identity element for addition, r + 0 –> r. Onwards, introduce sqrt (-1) –> i which allows every quadratic to gave a solution, and we see that i*i*r –> -r, and by extension i*r is orthogonal to the reals line. This gives us a 2-d planar space, with z = a + i*b as general co-ord of a complex number. In effect we have vectors that bring in rotations etc and can be handled algebraically with astonishing powers. we have reals as x axis, imaginaries as y axis, and space pops up. Extend to ijk unit vectors and a 3-d algebraically accessible mathematical world is there. Set trajectories based on x(t), y(t) for time and ideas of particles, inertia, momentum, energy etc and physics walks in. Indeed this is akin to how computer sims can be done (with of course discretisation and rounding.)

Is that mental world of forms a non-being?

Not if it has power to speak into the instantiated world as we see around us.

Are things such as the continuum or even the natural numbers, with all the infinities associated, nonsense and irrelevance?

Nope, they are very relevant and they are a big hint. They point to our finitude and limits, and they point beyond finitude. We can only imagine, point to a trend and put in the triple dot ellipsis. But, we see the real world power of mental ideas that point beyond us.

Indeed, we see necessary beings.

Eternal, uncaused realities that are inherently mental.

Blend in that if nothing — utter non-being — ever was, such would forever obtain, as non-being has no causal powers. Multiply by that we just saw infinities of inherently mental things with power that shapes what is possible.

We have a mind-shaped shadow cast on physical reality.

An eternal, mind-shaped shadow, as necessity plus actual reality implies that something always was.

Eternal, creative mind, as candidate no 1 to cast that shadow.

We are looking at the shadow of God.

Again.

We live in a God-haunted world, that reminds us of its roots at every turn.

So, perhaps, just perhaps, these things point somewhere.

As for issues on transfinite cardinalities, we need to take the triple dots seriously. The process, taken stepwise in order, never terminates. Instead, we must see such as laying out the in-principle order that would sequentially give us a set. The set, identified, can then be understood as actually forever present in full, hence its scale or cardinality is transfinite.

And, by applying operations that are logically consistent to transfinite sets we can see transfinitely scaled cardinalities. Where, two sets of entities have the same cardinality if they can be matched, element by element.

Five fingers with five, etc.

At the bar of the transfinite, where we move beyond what we can fully effect (WIKI: >>numbers that are “infinite” in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite . . . >>, where >>The Absolute Infinite is mathematician Georg Cantor’s concept of an “infinity” that transcends the transfinite numbers . . .>> i.e. there are distinctions) but can indicate, we see astonishing properties reminiscent of quantum strangeness:

That is, understanding cardinality as correspondence and reckoning with the triple dot effect, we see that because the sets are transfinite, the full set of naturals may be matched with proper subsets, the evens and the odds. Indeed, this is the property of inexhaustibility plus set-matching that Cantor used to designate transfinite cardinality. Here, aleph-null. (I suggest here for an introductory read: http://www.ias.ac.in/resonance.....8-0068.pdf and here also: http://www.cogsci.ucsd.edu/~nu.....rgmtcs.pdf )

Is this a case of blatant self-contradiction?

Not if you and I reckon with the impact of the triple dots, this keeps on endlessly, and we can accept the full sets. From this, we may move to even higher order transfinites, which are beyond countability. Most notoriously, the continuum.

Enter, stage right, the hyper reals and their inverses, the all but zero infinitesimals.

And, with them, points as locations without extension. As, implicitly, we have been using all along. Thence too, non-standard analysis and Calculus. This last, the crown jewel of and magic key to the scientific revolution. A mathematical achievement.

Such then become abstract models, forms if we will, that cast a long, long shadow on the physical world.

Whatever else he is, God is a mathematician of the first order.

And so also, utterly logical.

BTW, Cantor strongly associated the Absolute Infinite with God.

But also, out there somewhere, Plato’s ghost is laughing.

And he reminds us, from The Laws, Bk X:

In short, Plato answers the evolutionary materialist case by making a cosmological inference to design, including in his argument the human experience of being a conscious, minded en-souled creature. On that premise he intends to found public morality, thence an objective basis for just law. Which is his main target here.

Such a conclusion and policy programme, we may indeed choose to reject, but we cannot justly ignore nor censor it out of our considerations by a priori imposition of materialism dressed up in a lab coat.

So, then, I would not be quick to dismiss the issues linked to the mathematics of zero, infinity and their near-neighbours, so to speak.

Hey, mon, let’s enjoy the twelve days of Christmas and the new year as it comes!

KF

Phoodoo #451

Yeah, you can compare the reals to the natural numbers. And you will find that there are more real numbers. The cardinality of that set is bigger. There is no way to put them into a 1-to-1 correspondence. Look it up if you don’t believe me.

No matter what list you made of the real numbers it’s possible to prove you missed one. It’s possible to find one that isn’t matched with a natural number. Since you can find one that isn’t matched the set of reals must be bigger..

You can look up the proof. It’s quite straight forward. And was done over 100 years ago. This is NOT controversial except on UD. This is not materialist dogma or related to ‘Darwinism’. It’s basic set theory. Get Irving Kaplansky’s book, there must be 1000s of used copies about. Or get it through inter-library loan. Actually any set theory book will do.

The set of real numbers cannot be matched up 1 for 1 with the counting or natural numbers therefore the sets are different sizes.

en.m.wikipedia.org/wiki/Cantor’s_diagonal_argument

Don’t argue with me, read up on the mathematics.

Ph, pardon, there is no first real, the set runs all along the number line on both sides of 0 without upper or lower limit. Even if we specify magnitudes only |r| then for any number beyond zero we please 0.B, between 0.0 and + 0.B we can always specify another one by some means of interpolation. Another astonishing property, infinite fineness, if you will; thus also inherent non-countability which is not a contradiction. That is a direct implication of continuity. And, continuity makes sense coming from other directions, a lot of sense indeed, hence algebra, calculus, analysis etc. Specify, y = 2*x^2 + 1, where x is any real number. This is abstract, a parabola, and the graph we may sketch is yes a representation but that does not mean that the mental construct has no reality or physical influence — we can use this to construct a mirror for a search-light knowing that it will convert the light from a bulb at the focus to a parallel beam (car headlamps and flashlights apply this), or use it in a telescope and be confident that it will focus light from the stars . . . as Newton did in inventing the reflecting telescope. And, along the curve of that parabola, it makes sense to speak of the slope of the curve, thence differentiation, and of the area under it too, or the volume enfolded if we revolve around the y axis, hence wine glasses. And more, ever so much more. KF

Jerad, actually, no. Infinity is ALWAYS controversial. Abstract inherently mental realities tied directly to infinities and their near neighbours are inevitably strange, challenging and open to controversy. What is happening here, is that people are expressing their inner questions, doubts and frankly suspicions regarding an elite academic culture that has repeatedly cynically manipulated and betrayed their trust. For case study no 1, try EUGENICS, that extension of Darwinism, and linked Social Darwinism, which cost dozens of millions of lives within the past century or so. They have a right to be concerned, even to be suspicious. There is a price to be paid when a profession repeatedly violates the ethics of duties of care to truth and to limitations of knowledge claims, and you are seeing here at UD the tip of an iceberg people like us have warned about. KF

PS: Here is useful reading: http://www.ias.ac.in/resonance.....8-0068.pdf

Kf,

I have no idea who this post is in reference too. I think at times your answers are a bit too obtuse to really lead anywhere.

Jerad,

If you say that the set of real numbers is larger than the set of natural numbers, then first you must give some examples of your set. I will prefer to use 1 million as the first number in my list, now what is yours?

And really, if you are going to argue a logical point, please stop trying to refer to some vague authority of, look it up online; online is not an authority on anything, any more than Cantor is. if his argument can’t hold up to logic, its not a valid argument.

Now, my set starts with 1 million and proceeds in increments of 72, what is the first element in your set of real numbers?

Kf,

Do you wish to take a stab, what are the first two elements of your set of real numbers?

Kf,

I ask this because does not Cantors proof rely on the set of natural numbers being placed in some order. If that requirement is demanded to be offered in one set, then to be consistent it must be presented in the other.

The only way Cantors argument makes any sense at all is if we allow him to make the rules, and insist that the natural numbers follow a predetermined order, but the real numbers have no such duty. Why must one follow Cantors rules?

Can you answer that with words and not just symbols?

Phoodoo,

I’m not going to try and list the real numbers because i know it can’t be done ’cause no matter what list you give me i can always find one you missed.

I’m not suggesting ‘online’ is an authority. I am suggesting you can find pertinent materials, resources, information, discussions and examples online. I am saying you don’t have to believe me, you can find answers and other people’s work if you really are interested.

If you just want to argue with me then please learn the math you want to argue about. Find out what’s already been worked out.

Ph, The point is, in part that we cannot identify a first — or, perhaps, better, a

second— real. We can see 0 as first real, in a certain sense, but then any specific non-zero real, however close to zero, can be used to construct another value intermediate between it and zero. Above, I used the idea of place-value notation decimal numbers of form 19.78 etc which are WHOLE + FRACTION, I used W.B. In mathematical terms the non-zero reals are open intervals (0, infinity), ( negative infinity, 0) on the real number line. The round brackets mean, that there is no terminating value. Any specific value + 0.B or – 0.B will be such that no matter how close to zero, we can interpolate a value closer to 0. But, the details lie in university level Math. Sorry about that, I am not pulling rank, I am simply describing the level at which such is typically studied in technical terms. KFPS: Symbols are a technical language resorted to because they give enhanced precision. In fact, I am here fighting lack of access to graphical and non-Latin symbols in comments. Sorry, the apparatus is technical. However, in the above I referred to decimal numbers, the real number line and the like, that are accessible. I symbolise a general decimal number W.B for Whole + fraction, and assume understanding of the place value weighting.

PPS: I suggest a read here:

http://www.ias.ac.in/resonance.....8-0068.pdf

Jerad,

This is the whole point. You know you can’t show where I am wrong. You can’t list the first two elements of your set of real numbers. Until you can list them, how can you know that I can’t find a correspondence in the natural numbers?

You are trying to dodge the question, by saying it has already been answered, by someone, somewhere. Its typical materialist, scientific skeptic dodging. Its is very recognizable to anyone who questions Darwinists.

Kf,

I don’t mean to pull rank, but it is not a mathematical question, it is a logical one. You were not able to answer without a lot of symbols.

Its a simple question, why must the natural numbers be placed in order, to satisfy Cantors argument, but the real numbers must not?

Until you do a fair comparison of both sets, it is meaningless.

Ph, are you aware, that the glyphs used to represent words by chaining them are symbols with an associated calculable information content? That something is represented based on symbols does not count against its being meaningful or reasonable. I have pointed to primary school level mathematical concepts, which I think we can safely use, decimal numbers, real number line. Then, I pointed out that if you move away from zero to any specific number W.B (e.g. 19.78, or 0.00001978 for example) it will be such that you can ALWAYS construct another number closer to zero. One easy way is, take the average, (p + q)/2. Here, let p = 0, and you will see that the average will be at the mid-point of the line of numbers connecting 0 and q. For 19.78, that mid point number is 9.89. For the second number given, it is 0.00000989, per simple calculation. So, the fact of there being no “second” real, away from zero, points to how such is transfinite, and in fact to how there is an uncountable, much larger number of real numbers than of natural numbers such as 0, 1, 2 . . . and yes, the triple dots for something left off point onwards. These things may be unfamiliar, and may seem strange but they were just next door to what we all did in elementary education. Just, the structure of such education, very wisely, deferred the mysteries until later on, as most people have no practical need to address them. But if you deal routinely with analogue vs digital electronic systems, you have need. My simple illustration in a first digital electronics class was that you must stand on the rungs of a ladder but can hang on to a rope anywhere along its length. Extend that to the ideal case and the rest follows. And if you think this one is hard to swallow, try out quantum mechanics — which happens to be the best empirically supported physical theory, never mind the puzzles, paradoxes and more. KF

PS: Just to pick up one of Zeno’s paradoxical points, if the man running after the tortoise in a first interval covers half the gap, then in the second, half the intervening and so forth, he can and does overtake as common-sense observation shows. For each half-gap step takes a shorter and shorter time and within a finite time attains the limit of passing the tortoise. And so forth.

PPS, the issue is not a fair comparison, but to achieve an accurate and reasonable view. If it is so that we can demonstrate that there is no first non-zero real number, then we should accept that.

Kf,

You will not be able to understand Zenos paradox nor the problems with cantors argument if you insist that it is only a mathematical problem and not a logical one.

I understand that math is your field, so it is a disadvantage to you, to make you use thinking outside of your math training to answer a question. However, I suggest to you, that in order to debate someone, you will need to step outside of your comfort zone.

If the only way to accept Cantors argument is to apply rules to one set, that you don’t apply to the other set, you have not made a comparison. You are simply wrong.

The only way to apply Cantors logic is to distort what it means to compare. Thus the statement that one set contains more than the other is a non-starter until you realize that it is a logic problem, not a math one.

Phoodoo #461

That was your list of the real numbers? Add 72 each time?

You’ve missed out a lot of real numbers there dude.

Why should I make a list, you’re the one making the claim that runs counter to established results. Seems to me you have to prove your point.

If you think you can find a 1-to-1 correspondence between the reals and the naturals numbers by all means have a go. I don’t think you can so why do I need to come up with a list? I know it’s not possible to do so. Any list I make I can prove is wrong so why would I bother?

You think you can then go ahead.

I’d start with zero if I were trying. Then I’d try to list the infinity of numbers between zero and one. I’d try and find a scheme where nothing was left out. See if you can make that work.

Of course the problem is anytime you tell me there are two reals with nothing I between them I’d just find one. The gaps get smaller and smaller indefinitely.

You do know what the real numbers are? You do know that sqrt(2) is a real number. And pi. And e. Stuff like that.

phoodoo @ 464

Where did you read Cantor’s proof ? You seem to have a lot of wrong notions about Cantor’s proof.

Here are the basics (includes Cantor’s proof) in simple form. You can see that Cantor’s proof is logical. In fact since infinite sets can’t be written out, the only way to show it is not countable is by logic ! It doesn’t have any heavy maths at all.

Jerad,

Does my set of natural numbers have to be in order, or not? If they do, then so do yours.

If they don’t, then let’s keep it simple, what corresponds with the last of my numbers in the list of natural numbers?

If you want to say that there need not be rules applied equally to both sets, then my set is bigger, because it contains fairies, that yours does not.

Me Think,

Try to think, don’t just let others do it for you.

What are the rules for comparison between the set of real numbers and the set of natural numbers, are the rules applied equally? If they are not, then my set is bigger, it contains fairies.

What Cantor has done is a simple card trick. Anyone can play it. The object is to name a card that is the same as the other players card. Whoever can name the most cards the same as the other player wins.

But there is one catch, you have to always name your card first, then I name mine second. I will always win.

The question is, why does Cantor decides who lays down their card first?

How about we alternate who goes first, so that it is a real comparison, ok, I will start, 11, what is your match?

Now when we play a fair game of comparison, no one wins. It will always be a draw. No matter what you name, I can match it, and no matter what I name you can match. Its a draw! Pretty darn simple for anyone who can think without a calculator.

Phoodoo #467

They do not have to be in order as long as you can keep track of them.

There is no last number in your list. If you’re talking about starting with a million and keep adding 72 indefinitely

I expect you to use terms in the same way as mathematicians do or give a good solid definition of the terms you do use so I know specifically what you mean.

Anyway, you seem to be missing the point: you think you can demonstrate a 1-to-1 correspondence between the reals and the natural numbers. You haven’t yet shown me your matching scheme between the two sets. Or given me hint of what it might be. You’re talking about a set of integers for some reason. If you’re going to have a list of reals then sqrt(2) had better be on it.

If your going to make an argument that’s fine but you haven’t yet. This isn’t rocket surgery but it’s not a parlour game either. If you’re not going to take it seriously then don’t bother..

Jerad,

I can match any element in your set with an element in my set, if you think I can’t just tell me one element in your set, I can’t match from mine? Its that simple.

Phoodoo,

I’m sure you could. But that’s just showing a 1-to-1 correspondence between things I pick and your set. I believe you can do that. But that doesn’t show a 1-to-1 correspondence between the reals and the natural numbers.

What if my first number was the cube root of the imaginary number i? What if my first number was e^(-3i)? What if my first number was 428 mod 7? What if my first number was aleph-one? What if my first number was 0/0? Are you just going to say:add 72 to it, that’s my number?

That’s not showing a 1-to-1 correspondence between the reals and the natural numbers since I can’t list all the reals (as I said) which means we’d miss a real number just as I said.

What would e^(-3i) + 72 be anyway? Do you know? Is it in your set? Is sqrt(2) + 72 in your set?

No Jerad, you seem confused. I said I can match any element in your set with an element in my set, and still have some leftover. Its no different than what Cantor claims. Cantor simply plays a card game by saying the naturals must reveal their order first, then allow him to counter with his owned preferred one to one match.

If I turn it around and say the reals must reveal their set first, and I get to match up one to one, Then I win the game.

Cantors law only works if he games the rules.

Ph, logic is not only a branch of Phil, it is also a branch of Mathematics. And, the point of an open interval is that there is no specifiable first element. I have already shown that once you move away from zero, you have a case that for any given element, we can specify another closer to zero. Which of course feeds into the Zeno type issue, and through the physics of a series in time, we see why there is in fact a logical-mathematical basis for the easily observed fact that Achilles overtakes the tortoise in a calculable interval. In effect the additional increments of time and associated space converge to a definite finite time and space point (as each successive increasingly small increment of space takes a correspondingly small successive increment of time), the place and time of overlap, and after that A is ahead of the tortoise. As is familiar from simple observation. KF

PS: Applied Physicist actually, though I did pick up a third major, in math, in my u/grad work.

PPS: For the naturals, start with ordinals, and I already outlined the process that naturally yields the set as a set-builder process. To move to naturals, roughly speaking toss away the sequence ordering ops, to define cardinality. Thence, to get to the rest define operations etc as done in primary school, and get to the decimal numbers. That puts us in the space of reals once we see that we have whole number part and fractional part and can extend through addition so we have the place value notation representation of what is in effect an infinite series. Once there, we see continuity emerging and as a consequence, there will be no first real different from zero, we have an open interval. As was already pointed out in outline, I am just summarising. And, symbols allow us to compress the process, which in words is much like the descriptions of the Temple and Tabernacle in the Bible. Unwieldy and confusing because the flood of words gets us lost in the details.

Kf,

Then conversely mathematics is a branch of philosophy, so the philosophy must first be correct, for the mathematics to work.

If there is no first element in your set, then there is no first element in mine. We are at a stalemate. Cantor loses.

Are you sure you understand Zenos paradox of Achilles? The paradox is that he never passes the tortoise, if we stack the rules so that he can only cover half the distance each time. Likewise, Cantor’s “paradox” only works if you stack the rules.

Jerad:

I have, you ignored it and prattled on as if you haven’t been refuted.

Two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set. And that also means there isn’t any real 1-to-1 correspondence.Deal with that or you will prove that you cannot deal with reality.

With your example I have found an there are an infinite number of unmarried elements and using the mathematics of sets we have found them. That is by subtracting all the elements of one set from the other we, together, have demonstrated there are infinite elements left in one set but none the other. That means there are infinite elements left unmarried.Everyone sees that you are afraid to deal with that, Jerad. Thank you.

Jerad:

WRONG! 3n-1, as I have explained to you already. It’s as if you are unable to comprehend what I post.

You have nothing but what is being debated to support your tripe.

Infinity is not a number. It’s as if you are unable to think for yourself.

phoodoo @ 475,

Zeno’s paradox is akin to geometric series: sum of (1/2)^1 +….(1/2)^inf = 1, the sum converges, hence the distance can be covered by Achilles. There is no ‘stacking the rules’.

Mapou,

Maybe we’re not as far apart as I thought, then. Here is what I surmise from your post:

1) Pi exists as a real number.

2) You can estimate pi to any degree of accuracy you wish (presumably using a series representation?)

3) In fact, once you’ve chosen a series representation, if you choose a small number ‘tolerance’, there is a natural number k such that all the partial sums s_n with n > k are within ‘tolerance’ of pi. In other words, the sequence of partial sums converges to pi.

But then, by definition, we could write pi

equalsthe original series representation.I wonder if that is still acceptable to you, as earlier in the thread you stated that no infinite series exist.

MT, of course, the trick is the time steps drop with the distance steps and the infinite sequence descends to the infinitesimal, leading to the limit at the overtake point. Zeno “missed” that part. KF

phoodoo:Does my set of natural numbers have to be in order, or not?No, but you have to show that you have accounted for every element of the set. So if we take the rational numbers, we can show a one-to-one correspondence with the natural numbers. If we take the real numbers, we can show that no such one-to-one correspondence is possible.

Infinity doesn’t work the same way as finite numbers. A good example is Hilbert’s veridical paradox of the Grand Hotel.

Consider a hotel with an infinite number of rooms and all the rooms are filled. A new guest arrives. We can move the guest in room 1 to room 2, the guest in room 2 to room 3, the guest in room N to room N+1, and so on. Now room 1 is open for the new guest. We can do this again and again. The Grand Hotel is always full, yet always has room for more.

By the way, how many natural numbers are there?

DS, yup, once we move from the abstract domain of mathematical forms to calculation, we move to discrete approximate but useful values applicable to the physical world. As was noted hundreds of comments ago. Thus, we duly note the differences and proceed. KF

PS: in so proceeding, let us note that decimal, place value notation numbers such as 1.978 or 197.8 or 0.000001978 etc are all handy sugar-coats for INFINITE POWER SERIES on powers of ten, in form W.B. Such are in principle — beyond some partial sum and a relevant delta neighbourhood, going to be within that delta of the targetted real value, convergence guaranteed. Yup, infinite series sitting next door to grade school math. And with digital signal processing and Z transforms begging to come in the door also. (With the neighbouring Laplace Transforms in the briefcase.)

Ph, yup, EVERY intellectual domain targetting knowledge fits in with philosophy, the love of wisdom with particular focus on hard questions. Newton’s major work was in effect Mathematical Principles of Natural Philosophy. And the Ph.D degree boils down to, teacher of the love of wisdom. What we do is to hive off a new discipline when it seems reasonable to assign it a specialisation. But the meta-issues that come in with phil and issues such as logic, epistemology, metaphysics, ethics and so forth never actually go away. They tend to resurface when a paradigm is running into trouble, especially, and that causes a lot of trouble for those unaware of the issues that come with them or the required special approach of comparative difficulties. Hence BTW the significance of Phil-of-X meta-disciplines, such as for X = Science, Math, History, Religion, etc. KF

PS: The logic involved in the man chasing the tortoise, is that we have a double-acting series, in which both spatial and temporal steps are converging on a limit. As you get more and more shorter steps, you do them just as much faster and faster, so the tortoise gets overtaken at a predictable finite point in space and time. The calculus-algebraic expression will be abstract, and the series expression will converge on the same value. Reading (HT BA77):

Wolfram:

http://mathworld.wolfram.com/ZenosParadoxes.html

IEP:

http://www.iep.utm.edu/zeno-par/

Phys Forums debate:

https://www.physicsforums.com/threads/does-calculus-fully-solve-zenos-paradoxes.613003/

daveS:

Certainly not. 3.14 is a number. Pi is a symbol that represents a principle, function or calculation. I have always felt that Pi should be written with a different notation to indicate how many digits one wants after the decimal point. For examples, Pi-2 or Pi-16, etc.

The key concept here is this: “if you choose a small number ‘tolerance’”. And it is not a matter of if, either. One has to choose, otherwise nothing can be computed.

Not really. I repeat. There is no such thing as a truly smooth circle, a million fruitcakes claiming otherwise notwithstanding.

LoL! Infinity plus 1!

Zacky-O, the demon possessed fruitcake:

Zacky-O is the biggest fruitcake on this thread and probably every other thread as well.

kf @ 463:

(1) there are just as many even numbers as natural numbers!

(2) there are just as many natural numbers as rational numbers!

(3) there are more irrational numbers than rational numbers!

(4) there are more subsets of natural numbers than natural numbers!

I like this one. Check out the “More on Irrational Numbers” box at the following address:

The Real Number System

Mapou,

I guess some differences still remain.

And I can see the logic behind this notation, but what about cases such as the formula A = pi * r^2? Doubtless no exact circles physically exist, but we often do need to estimate the area of an approximately circular region. Or even find the radius of such a region given its area. I’ve always just used the symbol pi on its own, and have never reached the wrong answer to an applied problem by treating pi as a genuine real number.

Doesn’t that indicate that treating pi as a real number is, at the very worst, just a convenient shortcut? And that perhaps we should reconsider what we mean by “number”, possibly including things such as pi which can be expressed as the limit of a convergent sequence of rationals. Seems to me to be a very pragmatic step.

The real numbers have the property that they are ordered, which means that given any two different numbers we can always say that one is greater or less than the other. A more formal way of saying this is:

For any two real numbers a and b, one and only one of the following three statements is true:

1. a is less than b

2. a is equal to b

3. a is greater than b

Me_Think then:

Me_Think now:

Mung now: LOL!

daveS:

I have no problem with this, provided that one realizes that there is no such thing as a true circle. There never was and never will be. Sooner or later, Pi must be substituted with an actual value to obtain some result. I use Pi all the time in my speech recognition research. But my computer is finite just like everything else and I have to use an actual floating point number that is good enough for my calculations.

Joe,

Let’s try this. You have two sets

The natural numbers N: {0,1,2,3…}

Those numbers squared ‘N^2’ :{0,1,4,9…}

And I take it your argument is that that because there is a subset of N not present in N^2 then N must have a cardinality Card(N^2) + Card(N \ N^2) which is > Card(N^2) as long as Card(N \ N^2) > 0?

Here’s the problem: the cardinality of N, N^2 and N-N^2 are all the same,

and so is the their sum.Any infinite subset of a the natural numbers has the same cardinality as the naturals. We can show this by creating a “general” one-to-one correspondence:

For any subset “S” of the natural numbers, there will necessarily be a minimum value. We can map that value the lowest natural number. If we call the mapping function `f` then f(0) = s_0. The remaining elements in S must also have a lowest value, so we can keep going, mapping the lowest value in our subset to the next natural number:

f(1) = s_1, f(2) = s_2 …. f(k) = s_k.

That obviously maps a natural number to a member of the subset. The same relationship also uniquely maps any member of the subset to a natural number: for any k there are only a finite number of values

lowerthan s_k (there are k), so, the third non-square number maps to ‘3’ in the naturals and the k-th number uniquely corresponds to k.So, for any subset of the naturals, there is always a unique way to map a value in N to one and only one value in that subset. That is, they have the same cardinality, which is called aleph_0 or “countable infinity”.

Now, what does that do to your argument. You are saying

Card(N) = Card(N^2) + Card(N \ N^2) > N^2

But, if we put values we see that equation winds up telling us

aleph_0 = aleph_0 + aleph_0 = 2 * aleph_0

So, what you’ve actually shown is that 2*aleph_0 = aleph_0, which is trueand can be shown by the same “well ordering” trick above. In other words, you approach shows us one of the counter-intuitive properties of infinite sets, and helps establish that the cardinality of these specific (and in fact any infinite) subsets of N have the same cardinality as N.

Mapou,

Fair enough then.

Mapou, there are perfect circles, they are specified by r^2 = x^2 + y^2 and related operations. Just, they are non-physical, as in mental. KF

PS: I have already laid out how to get as far as the xy plane, etc.

wd400- If cardinality pertains to the number of elements in a set, then two sets cannot have the same number of elements if one set A) contains all of the members of the other set AND B) also contains members not found in that other set. And that also means there isn’t any real 1-to-1 correspondence.

The relative difference in cardinality between the two sets would be n squared.

Truly magical. Unfortunately magic doesn’t belong in math.

Mung, 488: Yup, there are a lot more subsets of the naturals than there are naturals. As in the cardinality of the set of subsets of a set of cardinality n is of scale 2^n. That gives us aleph_1 in this case, which is often held to be the continuum number but that is a hypothesis. KF

Joe, you are thinking in terms of sets with a definitive terminus, i.e. finite sets. Here, we deal with the transfinite. KF

PS: It may be helpful to make a loose analogy to order of magnitude. Naturals, evens and odds are of the same order of transfiniteness, but 2^aleph null is of exponentially higher order.

It occurred to me that the reason that Darwinists love this infinity crackpottery so much is that they need it to support their ‘infinite monkeys banging on typewriters’ hypothesis. After all, the only argument (assuming stupidity is an argument) they have to counter the obvious fine tuning of the universe is that there must be an infinite number of parallel universes and that we just happen to be in one that is well suited for life.

It’s truly pathetic.

KF:

I have to disagree. Nobody can imagine a perfect circle since the brain is finite. So a perfect circle is bogus regardless of how one defines it.

Having said that, given that some of you believe that the brain is not used for thinking, I can understand your position. But, IMO, it’s mistaken.

kairosfocus- Please see comment 218. I believe my claim encompasses the transfinite.

Sorry Mapou, but their theory requires

actualmonkeys.Mapou, coordinates of points satisfying the circle eqn are in a perfect circle, tied to Pythagoras’ theorem. KF

Joe, the sort of count you specify is a supertask and cannot complete as a transfinite cannot be traversed stepwise. You have to deliver the sets all at once or else have the sort of speeding up Zeno implied that completes the series in a limited time. The transfinite is tricky, at any given finite time a double-fast count will exceed the base count, but you have not even approached the transfinite yet. That is why one looks at set-builder logic and compares the scale which as shown yields that say 1, 2, 3 . . . x2 gives 2, 4, 6 . . . less 1 gives 1, 3, 5 . . . and THAT gives the same order of scale. To break out beyond that you need to go to exponentials such as the power set which goes to 2^aleph_null. KF

“The introduction of set theory at the end of the nineteenth century persuaded many mathematicians, Bertrand Russell among them, that they had discovered a system by which the natural numbers could be displaced in favor of something more fundamental. The creation of Georg Cantor, set theory is the most remarkable single achievement of nineteenth-century mathematics, so much so that David Hilbert was moved to call it a paradise.”

– David Berlinksi,

One, Two, Three: Absolutely Elementary MathematicsKF @504,

Again I disagree. If it uses numbers, it is discrete. I will not argue the point beyond this, sorry.

Mung @506,

As we all know, mental illness runs in David Hilbert’s family. “Fruitcakism” is alive and well in mathematics and science.

Mapou:If it uses numbers, it is discrete.A circle (at the origin) is the set of

everypoint on the plane that satisfies the condition r^2 = x^2 + y^2 in Cartesian coordinates, or simply r = a in polar coordinates.Demon possessed Zacky-O:

Sorry. A perfect circle is not a formula. It’s an idealized construct that can never exist.

Another example of a fruitcake is Stephen Hawking, the crackpot in the wheelchair whose former wife once said that her duty in the marriage was to remind Hawking everyday that he was not God.

Against all logic, Hawking insists that time travel is a possibility in spacetime or “Einstein’s universe”, as he calls it. And yet, anybody with less than two neurons between their ears knows that spacetime is a block universe in which nothing happens. This is something that did not escape the great Karl Popper who wrote about it in “Conjectures and Refutations” and compared Einstein to Parmenides, the Greek philosopher who denied change and motion.

PS. In my opinion, Parmenides (whose work we know by way of Zeno and Aristotle) did not really deny motion/change. He was just trying to point out that there is something fishy about continuity. But I digress.

kf @ #496 & Mapou @ # 501,

I have to disagree with you both. There is one ideal circle and it can be perceived by the mind. The ideal circle does not become a multiplicity just because it is apprehended by several minds. The ideal circle in my mind is the same ideal circle that is in your mind.

Kf,

All you have to do is name the elements in your set, and I can show you a one to one correspondence from mine.

The problem is not in finding the one to one correspondence, it is in you naming the elements of your set. Give them a name, and I can show you a correspondence in mine.

Can you name Two that don’t have a one to one correspondence in mine? Ok, if you can’t name two, then how about One?

Cantors is not a mathematical proof, it is a slight of hand mental card trick. It is because many mathematicians are caught off guard having to defend words, that we end up with many of the poorly solved realities in the world that Mapou has mentioned.

Until you can name One element in your set, that I can’t find a one to one correspondence with from mine, Cantors work is still just that of an amateur magician in history. You have explained no realities in the world.

Joe.

Truly magical. Unfortunately magic doesn’t belong in math.What’s wrong with the math that showed it be true?

Can you define a subset of the natural numbers that can’t be put into one-to-one correspondence with the natural numbers?

phoodoo #473

Sorry for taking so long to catch up. It’s been a long day

Look, you are making a claim that YOU can find a 1–to–1 correspondence between the reals and the natural numbers. It’s up to you to defend that claim.

I am absolutely NOT going to play that game because I think it’s not possible to find such a correspondence.

You can’t claim a victory because you devised a personal version of what you think Cantor was saying.

Read his argument why there are more reals than natural numbers, it’s easy to find online, and then lets talk. I’m interested in particular criticisms you might have of his method. Please be specific.

But, as I already said, I don’t think it’s possible to list all the real numbers. Either explicitly or implicitly or recursively. I don’t understand you argument to be honest. You’ve devised your own test. arguing against a strawman which no one is defending.

Cantor says: lets say I can create a recursive list of all the real numbers. It’s a list so I can number it. that gives me a 1-to-1 correspondence with the reals. BUT I can create a real number which is NOT on the list. So the list wasn’t complete. Add that number to the list. And then create another number not on the list.

This is the real point. With countably infinite sets you CAN create a correspondence with the natural numbers. Anytime you try and do that with the real numbers you find a real number which is not matched up. Therefore, the set containing all the reals is bigger than the set containing the natural numbers EVEN THOUGH both sets are infinite.

Ph, every decimal numeral is a name. And, an existence proof is a proof. KF

Mapou and CM, the set builder criterion has specified the circle. And as r may take a considerable and even continuous value of ranges, the family is indefinitely dense. KF

Joe #476

And yet I have come up with a 1-to-1 correspondence that you have been unable to find a counter example.

In my example of {1, 2, 3, 4 . . . } and {2, 5, 8, 11 . . .} wherein I link an element n in the first set with an element 3n – 1 in the second set, every element of both sets is accounted for and linked to an element of the other set. Unless you can find an unlinked element of either set.

I have been doing so. I found a matching/mapping that does what you claimed can’t be done. Why don’t you show me where I am wrong?

Tell me one of your unmarried elements from my sets {1, 2, 3, 4 . . . } and {2, 5, 8, 11 . . . } using the matching n in set 1 matches to 3n – 1 in set 2. Tell me an element, a number, in either set that does not get matched to a single element of the other set. No hand waving, no bluster. Give me an example.

And what is that size difference Joe? And how do you get that from the formula that I provided? Be specific.

Unlike you, I will not speak for everyone. But I do await your providing the examples you say exist.

Joe #477

I am sorry if you haven’t been clear. What exactly does that formula tell you about the cardinality of the sets involved? Please be specific. And say how you draw your conclusion from the formula WHICH I PROVIDED. Not you.

Mapou #511

Someone please tell me that I am not the only participant who finds this offensive.

Are there no moderators on this thread?

Jerad:

What’s offensive is being lied to by famous people with access to a bully pulpit. It’s not nice to deceive the public. I, for one, am deeply offended.

PS. Why is Hawking revered as a great physicist when he does not even know that Einstein’s spacetime is a block universe? And this is his specialty.

phoodoo #513

This is pretty much what Cantor did. He said: create a list of all the real numbers and I will link them to the natural number simply by numbering your list. But, I can find a real number that is not on your list even though you think you listed them all. If you then add that number to your list I will create another real number not on your amended list.

You can list, projectively, all the natural, counting numbers. If you try and do the same thing with the real numbers a clever observer will always be able to find one you missed.

That’s the argument.

Mapou #521

If you’ve got a serious and credible argument against what Dr Hawkins is saying then by all means publish it. But do not slander people just because you disagree with them

KF has taken a strong stance against slanderous comments and behaviour. I wonder what he thinks of your statement.

Jerad:

You are the one who is slandering me for accusing me of slandering Hawking. What I said about Hawking and his time travel crackpottery is the truth and I can prove it. And yes, I have published my opinion of Hawking on my blog. Sue me if you take offence.

Set 1 = {1, 2, 3, 4 . . . }

Set 2 = {2, 5, 8, 11 . . . }

Match an element n in set 1 to an element 3n – 1 in set 2. That means that the element 1 in set one gets matched to 2 in set 2. 2 in set 1 gets matched to 5 in set 2. 3 in set one gets matched to 8 in set 2. etc.

This proves that I can match every element of both sets uniquely to an element of the other set. Tell me an element of either set and I can find its match in the other set. And I further say that this can only be true if the sets are the same size. And I concede that if you can find an element of either set which is not matched up with my criteria then I will have been proved wrong.

So, can you find an element of either set that is not matched up under my criteria?

That’s it. Have a go as we say in England. You find an unmatched element under my scheme and I’ll concede.

Can we just concentrate on this one simple case?

Mapou #524

I can’t sue on behalf of Dr Hawking. And I do applaud you for having the chutzpah to stand behind your opinion. At least you’re not some one who takes a shot and then runs.

Whether or not what you claim is true really is true; I shall leave to the physicists. I suspect you are wrong. But I will defend to the death your right to say what you think. I think we can agree that that right matters. A lot.

Jerad:

LoL! Mine was the counter-example to yours and you have not been able to counter that. I will not continue to refute imaginary matches. The only way you will ever prove your case is to show a practical use that could only exist if it were true. Something like A^2 + B^2 = C^2.

Mathematics is a useful concept and it isn’t philosophy. Bad things tend to happen when we get the math wrong.

Will trajectories be affected if we all agreed that the cardinality of countably infinite sets could be different? Would relativity be refuted? Would the climate ease to change?

What calculations would be affected?

That is if we allowed the standard and accepted definitions and rules of set theory to hold universally, as universal laws do (see the math), meaning ALL proper subsets will have fewer elements than their “parent” or super set, what, in the real world, would be affected? What would be the effect of such a thing?

We could then use the “mapping formula” to tell us the relative difference. For example:

Let set A = {1,2,3,4,…}

Let set B = {2,4,6,8,…}

Would be a relative difference of 2n, meaning set A is twice the size of set B.

And one more time- If Jerad’s methodology was correct I would not be able to use standard set mathematics to prove him wrong. And yet I have.

Jerad:

AGAIN- IT’S YOUR SCHEME THAT IS BEING DEBATED. That means you cannot just keep trotting it out as support for itself. And you cannot justify using your scheme for the reasons provided.

Is this not getting through?

Can anyone define a proper subset of the natural numbers that when each element is removed from the set of natural numbers, leaves an empty set?

If you can I will accept that set/ those sets and the set of natural numbers has/ have the same cardinality.

Short of that all you have is pure trickery and I am calling shenanigans. 😎

Mapou:A perfect circle is not a formula. It’s an idealized construct that can never exist.A circle is the set of points equidistant from a center point.

phoodoo:All you have to do is name the elements in your set, and I can show you a one to one correspondence from mine.The real numbers.

Jerad:

90% of what makes good science is guts, IMO. I have no fear; so I have an advantage over others. It’s too bad science has been invaded by cowards, butt kissers, crackpots and liars. Fortunately, it will all come to an end soon enough. Wait for it.

This is because you are a crackpot just like Stephen Hawking. I gave the proof of the impossibility of motion in time or spacetime in this thread. It’s not rocket science. So I can’t say that it went over your head. I can only conclude that you, too, are a coward.

I would stand my ground even if I did not have that legal right.

nothin’ from nothin’ leaves nothin’

ya gotta have sumthin’

if ya wanna be with me

Right. An infinite set of points.

Thus nonsense…

In any

realsense.In reality it boils down to precision.

No such thing as infinite precision…

In reality

Right.Wheels don’t exist ,in any real sense. Wheels are figment of your imagination.

Here’s a question for all those who think the idea of an ideal circle is nonsense. What exactly

isthe set of all (x, y) in the plane such that x^2 + y^2 = 1? Certainly it’s a nonempty set, as (1, 0), (0, -1), (8/17, 15/17), and a bunch more points satisfy the condition x^2 + y^2 = 1.If you plot all these points in the plane, they seem to lie more or less on a circle of radius 1.

If it’s not an ideal circle of radius 1, what is it?

Mung @ 492

Just saw your comment . I don’t see any logic in

your LOL.The 2 statements are not contradictory at all.Me_Think:

LOL. So you counted all the points on a wheel and you found them to be exactly what? Infinite?

What a bunch of geniuses you Darwinists are. We’re just a bunch of morons compared to your infinite wisdom. I’m impressed. Very impressed.

Mapou, I think you need to dial back barbed verbiage and name-calling. Insofar as you have a case on the merits, make it on the merits. KF

KF @538, will do. Not because I apologize to anybody but because this site does not belong to me. And I will respect that.

Folks,

I think one of the problems is failure to distinguish establishing the logic that builds a set or matches (or fails to match) members of distinct sets in 1:1 correspondence, with a stepwise exhaustion of the members.

The former is achievable for transfinite numbers, the latter is not.

But, if you are willing to go with the force of logic, the former should be enough on pain of good old selective hyperskepticism. Further to this, in dealing with abstract mathematical objects, we are dealing with a mental, abstract world of forms, variables and the like.

That such a world can be shown to have sufficient reality to influence what happens or can happen in the real world — e.g. no square circles can exist — should give us all pause.

When we turn to physical implementations such as forming a wheel or the disk that we intend to cut gears on in light of a mathematical specification, we cannot fully implement the ideal, not least as there will be atoms distributed in metal crystals, and at practical level it is hard to do things to a precision of 1/1000 of an inch, consistently. (Oops, there was a key skip.)

Ask any machinist, or ask the engineers at Abu as to why the Record reels of the 1940’s – 50’s did not have interchangeable parts. But by 1954 or so, the new Ambassadeurs, did. Likewise parts of the old 0.303 SMLE were not interchangeable, e.g. even though the magazines were detachable, they were specific to a particular rifle (and in early models were chained to it, I gather). That’s why loading was by 5-round charger clips. I don’t know if the later Ishapore India Rifles, now chambered in 7.62 NATO, are fully interchangeable, but on general progress of tech, I suspect so. (Yes, the SMLE — India and Australia retained the WW I model — is still a police rifle, and may hold reserve status in India.)

But, surprise, the mathematical world is able to address these issues, and that is how later models were made with sufficient precision to have interchangeable parts. Hence, statistical process control and quality management.

We are back at Wigner’s remark about the astonishing powers of Mathematics in the physical world. And the implication that logico-mathematical constraints and opportunities were built-in from the outset. Pointing in very interesting ontological directions.

Coming back, the key point is, are we willing to go with the force of the logico-mathematical points when they are sufficiently shown?

If so, then if we can show how 1:1 correspondences exist with transfinite sets and subsets through transformations such as:

. . . then we should accept that.As, here, patently, we have transformed the full set into subsets. Even, when we can also see that trying to subtract elements is problematic. This may seem strange, but the force of the logic is, we cannot stepwise traverse and exhaust the sets, which is a situation that goes beyond the world of our common experience.

We are forced to address pure logic, and it gives unpalatable results for the abstract world of mathematical forms. But, without that world, we lose the logical foundations of Calculus and wider Analysis, with devastating impact. For, calculus was and remains the crown jewel and magic key of the scientific revolution.

That’s what we are playing with here.

KF

Joe, various

I am not denying your matching gives a different result but you can’t say yours is right and mine is wrong without finding something wrong with mine. Can you find an unmatched element in either set in my matching? I don’t think you can since you haven’t come up with one. If all elements of both sets get matched up with elements from the other set then the sets must be the same size. This is what you have to focus on.

Nope, reality doesn’t care about what we think.

Can you find an unmatched element in my scheme or not?

If I’m right then it is possible for a proper subset to have the same number of elements as the parent set. Welcome to the infinite world.

Except that you can match up the elements 1-to-1 so the sets are the same size.

No, you haven’t. You haven’t found a mistake. You’re just asserting what is true for finite sets. It doesn’t hold for infinite sets. Go look up that rule you keep quoting and see if it says for all sets or only for finite sets.

Don’t debate it, find a mistake. Find an unmatched element using my scheme. You keep using finite reasoning for an infinite problem.

You haven’t found a mistake in my matching. You haven’t found an unmatched element. Why don’t you admit that Joe? Your finite rule doesn’t work for infinite sets.

Joe,

Perhaps you would get the same result as me if you used a 1-to-1 matching like I did.

And keep looking for an unmatched element in my scheme. That would definitely prove me wrong.

And again, welcome to the infinite world. I love what KF said (I paraphrase): God is a kick-ass mathematician.

KF

Hear, hear!! Which has the interesting duel meaning of “I agree” and “listen, listen”.

Mapou @ 537

Pi is irrational and Real number and is part of Cantor’s set, yet it is used for real objects. The point is, it doesn’t matter if you think Cantor was a nut. There are irrational numbers in Wheel’s manufacturing.A wheel is a real object which is a circle – that’s all that matters.

MT: Pi is worse, it is transcendental, one cannot construct a polynomial eqn with rational coefficients whose root will be pi. To estimate it for practical purposes we are left with successive approximations by infinite length power series analysis, duly truncated on convergence. Or the like. And such brings us to the epsilon-delta analysis on partial sums. As in, beyond a partial sum of n terms, for some value of tolerance delta, the difference between the partial sum and the elusive final value will be some error epsilon that is demonstrably less than delta. But to get there all the stuff on transfinites, continuum, infinitesimals and more lurk. Not to mention that place value notation decimal numerals (or the equivalent for binary, hex etc) all are infinite series expressions in disguise, typically with suppressed leading and trailing zeros of indefinite but large number, in principle transfinite. Grade school math sits next door to the most profound mysteries. KF

@ Me_Think

“A wheel is a real object which is a circle – that’s all that matters.”

Obvious non-sense!

inunison @ 546

What is a wheel’s shape? Don’t know the answer ? Ask a kid.

MT, a wheel is close enough to ideal for govt work, but it is never quite perfect. There will be approximations boiling down to effectively a very high order n-gon. It’s like how monofilament line does not have a perfectly consistent shape. Perfect circles are ideal, logical-mathematical, not physical world. KF

Hi KF, you wrote in #517:

I’m not really arguing with you, just trying to clarify how we use the term, “perfect”. Your set of circles do not exist in the material world nor do they exist in the ideal realm. They are a mathematical construct or tool by which we can approximate to the one ideal circle, actual material circles or circles we have thought up . None of your perfect circles can exist in any absolute sense because they each have magnitude which is relative. The absolute, ideal circe is a unity which exists in and of itself and is not relative to anything outside of itself.

So your perfect circles do have a perfect aspect but they are not absolutely perfect.

Hi MT, as KF said a wheel is an approximation of a circle. No true circles exist in the material world.

CM, any circle has a specific radius, thus the circle is a mathematical ideal, there is no one circle. The function specifies any circle which exists as the instantiation thereof. KF

Jerad,

Right, Cantor said, you create a list of all the natural numbers FIRST, then I can tell you which of my numbers you haven’t found a match for. So I can turn it around and say the same thing, YOU create a list of all of your real numbers, and then I will simply show you one by one how I can match up a number of yours with a number of mine.

Since you can’t even start with two from your list however, I guess that makes it impossible-that’s the card trick. I can list two, and you can’t. Its pretty hard to draw diagonals on your list when you don’t even have one.

Jerad:

LoL! Obviously you don’t know set theory as a proper subset is defined such that what you say is impossible.

AGAIN- IT’S YOUR SCHEME THAT IS BEING DEBATED. That means you cannot just keep trotting it out as support for itself.And you cannot justify using your scheme for the reasons provided.LoL! Talk about being ignorant of how to debate something! I found the mistake, Jerad. And all you can do is use your mistake-ridden formula to “rebut” what I found.

So you say yet cannot prove.

And it is very telling that Jerad cannot show ONE practical application that refutes my claims and supports his.

kf, think of the essential nature of the ideal circle. The ratio of the radius to the circumference is unchanging, it is an absolute value. No matter what size a physical circles are they will all be an equal approximation of the ideal, everything else being equal. Magnitude is relative and relativity has nothing to do with the ideal. When defining a circle, there are no text books where size is considered. Size only comes into play when we are considering the physical world.

Mathematics is a useful concept and it isn’t philosophy. Bad things tend to happen when we get the math wrong.Will trajectories be affected if we all agreed that the cardinality of countably infinite sets could be different? Would relativity be refuted? Would the climate ease to change?

What calculations would be affected?

That is if we allowed the standard and accepted definitions and rules of set theory to hold universally, as universal laws do (see the math), meaning ALL proper subsets will have fewer elements than their “parent” or super set, what, in the real world, would be affected? What would be the effect of such a thing?

We could then use the “mapping formula” to tell us the relative difference. For example:

Let set A = {1,2,3,4,…}

Let set B = {2,4,6,8,…}

Would be a relative difference of 2n, meaning set A is twice the size of set B.

And one more time- If Jerad’s methodology was correct I would not be able to use standard set mathematics to prove him wrong. And yet I have.How did I prove Jerad wrong? By using basic set subtraction I was able to demonstrate That I can remove each and every naturally matched element and still have an infinite set left. And that Jerad refuses to deal with that proves he has nothing but to keep repeating his refuted tripe.

And I am OK with that…

Jerad:

I did. Yours is NOT natural. Yours is a false relationship. Also you cannot say that yours is right and mine is wrong without finding something wrong with mine as I did to yours.

Good luck with that.

KF @ 548

CharlieM @ 550

A perfect circle can’t exist in any world – not possible even in Mathematical world, since Pi can’t be computed fully ( as it is irrational). It is ridiculous to expect a perfect wheel. If there is a perfect circle, then Pi will become rational.

A perfect wheel will, in fact, be more bumpy than a ‘real world imperfect’ circular wheel, as we can never have a perfect plane. At microscopic level, even a ‘perfect’ plane is just a series of cantenary shape. The centroid of a perfect wheel will never follow a straight line if the wheel is a perfect circle hence the ride will be bumpy even if the road is a ‘perfect’ plane.

Me_think says,

A perfect circle can’t exist in any world – not possible even in Mathematical world, since Pi can’t be computed fully ( as it is irrational).

I say

So in your worldview existence is conditioned on complete computational reduction.

That explains a lot.

I hope you know that there are lots of worldviews that are not so constrained.

Lots of us have no problem with real things that can’t be computed. In fact some us us hold that the primary reality can not be computed.

quote:

Yes, may you come to know his love—although it can never be fully known—and so be completely filled with the very nature of God.

(Eph 3:19)

End quote:

peace

phoodoo #552

But it’s impossible to create a list of all the real numbers so why should I try?

Besides you’re the one making a claim that runs counter to accepted results. You make a list.

It’s impossible to make a list of the reals, that’s what Cantor said. I know that no matter what list I made you could easily match an integer to everything on the list but my list would always be incomplete. So, your game is a non-starter.

Any list of real numbers will be incomplete, that’s the point. That’s what Cantor showed.

Joe #553

From https://www.proofwiki.org/wiki/Infinite_Set_Equivalent_to_Proper_Subset

“A set is infinite if and only if it is equivalent to one of its proper subsets.”

Welcome to the infinite world.

You haven’t got any reasons. You said it was ‘contrived’ but you couldn’t back that up with a referenced definition. You don’t like my scheme ’cause it proves my point.

Find a documented reason why my scheme is wrong or questionable. You haven’t so far.

Besides, ti’s dead simple.

You found no mistakes. You said it was contrived but couldn’t back up that claim. YOU think the only matching that’s possible is one that matches up identical elements but that’s not true so it’s not a mistake.

I already have.

Funnily enough that doesn’t matter when you’re talking pure mathematics. Besides, I don’t bump into too many infinite sets in the real world.

fifthmonarchyman @ 558

Reality is different from an ideal world view. There can be no real worldview which is not constrained in some parameters.

Jerad, If I can remove all of the elements of the proper subset and still have elements left then it is obvious the two sets are not the same size. I am using normal set math. You are using imaginary relationships. That is the mistake I found, Jerad. Your willful ignorance is not a refutation.

And obviously you are also ignorant of mathematical proof.

LoL! Wrong again, Jerad. The only matching that counts is the natural matching. all others are manufactured and show a false relationship.

OK so there isn’t any practical application and that means anyone can say anything they want about this part of set theory and no one will be able to prove them wrong.

It is a sad day for mathematics.

And AGAIN:

Mathematics is a useful concept and it isn’t philosophy. Bad things tend to happen when we get the math wrong.Will trajectories be affected if we all agreed that the cardinality of countably infinite sets could be different? Would relativity be refuted? Would the climate ease to change?

What calculations would be affected?

That is if we allowed the standard and accepted definitions and rules of set theory to hold universally, as universal laws do (see the math), meaning ALL proper subsets will have fewer elements than their “parent” or super set, what, in the real world, would be affected? What would be the effect of such a thing?

We could then use the “mapping formula” to tell us the relative difference. For example:

Let set A = {1,2,3,4,…}

Let set B = {2,4,6,8,…}

Would be a relative difference of 2n, meaning set A is twice the size of set B.

And one more time- If Jerad’s methodology was correct I would not be able to use standard set mathematics to prove him wrong. And yet I have.

How did I prove Jerad wrong? By using basic set subtraction I was able to demonstrate That I can remove each and every naturally matched element and still have an infinite set left. And that Jerad refuses to deal with that proves he has nothing but to keep repeating his refuted tripe.

And I am OK with that…Except those two sets are the same size.

Nope, see the theorem (that means it’s been proved) I referenced above which says that a set is infinite if and only if it is equivalent to one of its proper subsets.

Welcome to the infinite world.

Because I found a 1-to-1 matching while your subtraction is not a 1-to-1 matching. It’s pretty simple really.

Not natural? What does that mean? You couldn’t back up contrived so now your saying my matching is not natural? It’s a false relationship? What are you talking about?

I didn’t say yours was wrong, it’s just not 1-to-1. There are lots and lots and lots of ways to match up the elements of those two sets. I found one that’s 1-to-1 which can only be true if the sets are the same size.

I really don’t understand this notion that there’s only one (natural?) way to match up elements of two sets? Can you please show me a set theory book or article which says that?

I don’t need luck.

Me_Think says

Reality is different from an ideal world view.

I say,

Correct, Your worldview determines what qualifies as reality

You say,

There can be no real worldview which is not constrained in some parameters.

I say,

What exactly is a “real” worldview?

Can a worldview be real if it can not be computed fully?

peace

Zachriel:A circle is the set of points equidistant from a center point.mike1962:Right. An infinite set of points. Thus nonsense…Heh. It’s the definition.

kairosfocus:Pi is worse, it is transcendentalWe prefer to think of it as

better.phoodoo:YOU create a list of all of your real numbers, and then I will simply show you one by one how I can match up a number of yours with a number of mine.As Cantor proved, no one can list the real numbers.

Joe #562

Nope. Take the natural numbers 1, 2, 3, 4 . . .

Tale out the infinite set 2, 4, 6, 8 . . .

You’ve got left 1, 3, 5, 7 . . .

All three of those sets are the same size. And I can prove it by finding a 1-to-1 correspondence between any pair of them.

You need to stop using finite notions and start using infinite ones.

I didn’t make a mistake, you’re not working with infinite sets properly. I am not dodging an issue,

Really? Well, the theorem I linked to had three proofs associated with it. If you can find a mistake in any of them I’ll reconsider.

A contention which you have not yet referenced or justified. Please do.

Not at all. A lot of people thought the Axiom of Choice was obviously true but it ain’t. That’s why it’s called an Axiom now, even though it’s equivalent to Zorn’s lemma which is confusing really. And it’s equivalent to the Well Ordering Principle. The names are historical hold-overs.

Look Joe, if you’re right then you should be able to find set theoretical definitions of ‘contrived’ and ‘natural’ regarding mappings (which is the way mathematicians referring to the kind of matchings I’m doing).

Think about a function . . . say f(x) = 3x – 1

Put in an x, say x = 3, get out a result, in this case 8. The function matches 3 with 8. AND, if you draw a picture (graph) all the pairs that the function ‘creates’ you get a straight line. A function is a matching between two sets, normally called the domain and the range. This function is 1-to-1 because every element in the domain (the x-values) is matched which exactly one result or outcome (the y-values when we graph the function on the (x,y) plane).

The function I’ve just listed is the same as the matching scheme I used between my two sets. My matching is a 1-to-1 function between the two sets. It tells you how to get from an element in one set to its unique counterpart in the other set.

Jerad:

Thank you for proving my point. Have a good day.

I am not trying to be belligerent nor am I making absurd claims. The mathematics I’m referencing and the techniques I’m using are not controversial or complicated. The theorem I linked to (A set is infinite if and only if it is equivalent to one of its proper subsets) is not new or made up. And there’s three proofs on the web page.

Anyone can look up set theory on Wikipedia, read and follow some links and find most of this out. It’s not just me saying these things.

I don’t understand why some people are having such a hard time with it.

fifthmonarchyman:Can a worldview be real if it can not be computed fully?There are infinitely many noncomputable numbers*. While we might speculate the universe is discrete and computable, that is not necessarily the case. Reality doesn’t have to stoop to our limited expectations.

* Computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers or the computable reals.

Jerad, If you were right then we wouldn’t be able to do what I said and still have an infinite set left. That proves that yours is a manufactured scheme meant to fool people.

Obviously it has worked on you and others. Don’t blame us because you are easily fooled.

Joe #568

And all three of those sets are the same size. Why did you leave that part off?

fifthmonarchyman @ 565

The world that one faces when s/he is awake and sober is the Real Worldview.

Any world view can be computable only up to a level that we understand it, so computability is progressive – we can compute more today than a few decades back, and we can compute more a decade from now.

Zac says,

Reality doesn’t have to stoop to our limited expectations.

I say,

amen

peace

Me_think says,

Any world view can be computable only up to a level that we understand it, so computability is progressive – we can compute more today than a few decades back, and we can compute more a decade from now.

I say,

Just like Pi

Some worldviews allow us to continue to progress in understanding infinitely.

Others are constrained to only understanding the finite phyiscal universe.

peace

Zacheriel,

If the real numbers can’t be listed, how can you make a one to one correspondence?

Joe #571

Huh? The natural numbers are countably infinite. The even natural numbers are also countably infinite. The odd natural numbers are countably infinite. You can take an infinite set away from another infinite set and still have an infinite number of things left. Obviously.

Have you looked at that theorem or found a reference for your use of the term contrived or have you found a reference for your claim that only the ‘natural’ mapping/matching is right?

I can’t blame anyone for finding dealing with infinities weird the first few times. Which is why I think it’s good to do some reading and see what’s already been done first. It’s easy these days!!

phoodoo:If the real numbers can’t be listed, how can you make a one to one correspondence?It’s provable that you can’t make a one-to-one correspondence between the natural numbers and the real numbers. However, it’s provable that you can make a one-to-one correspondence between the natural numbers and the rational numbers.

phoodoo #576

You can’t!! THAT’S THE POINT!! There are more real numbers than integers!!

MT @ #557:

MT, I don’t much care for labels, but: I presume you are a materialist. I would call myself an objective idealist.

If you think about it a perfect circle can’t exist in the physical world. From your comments you are giving us an example of a perfect circle in the physical world. Well no one is arguing that this is a possibility.

Why do you think that Pi needs be rational? It might conform to neatness and tidiness according to our human minds, but what has that got to do with anything? The fact that Pi stretches on to infinity is no problem for the ideal world. It is the world of infinities and absolutes.

You say the perfect circle can’t exist in any world implying that you know all worlds. The ideal circle which I say is real will be the same circle throughout all of time, its definition will not change, but any wheel you care to mention had a beginning and it will have an end. In other words my circle is permanent but yours is transient. But you still say that yours has more reality than mine. Obviously I disagree.

I’ve been following the Joe/Jared dialog.

None of Jared’s posts are his own personal opinion or unconventional. Every point he has made is standard mathematical infinite set theory that can be found on the internet or in any textbook on the subject.

.

As for Joe, so far I’ve been unable to discern whether

a)He is not familiar with (or doesn’t understand) infinite set theory,or

b)He understands infinite set theory but simply rejects it..

If it’s (a), that’s easily resolved by doing a bit of homework.

If it’s (b), then Joe is certainly free to make up his own mathematics. But it would be helpful if he would clearly state that’s what he’s doing, and rigorously define the terms he uses.

.

Joe’s theory:

Ok, so based on this, set A would be thrice the size of {3,6,9,12,…}, five times the size of {5,10,15,20,…} and 2.33333… times the size of {7/3, 14/3, 7, 28/3, …}.

Got it.

But wait – what about subsets that can’t be formed this way? No problem, says Joe – just take the function from which they’re derived, and divide by

n. Easy.So, for example, {1,2,3,4,…} is

ntimes the size of {1,4,9,16,…}. Joe himself confirms this:Except…

nisn’t a number. Saying one of these sets isntimes the size of the other makesno sense at all. Does their difference in size increase? Is it as large as you want it to be?? Could it be infinite???Further examination of Joe’s ‘theory’ produces more peculiar consequences. Presumably {1,2,3,4,…} is 2

ntimes as large as {2,8,18,32,…};n^2 times as large as {1,8,27,64,…}; and 1/nlarger than {2,3,4,5,…}.But what about these sets?

{1,4,27,256,…}

{1,1/2,1/3,1/4,…}

{1,sqrt(2),cbrt(3),sqrt(sqrt(4)),…}

Standard set theory has no problem at all, but Joe’s ‘theory’ explodes in a frenzy of leaking grey matter.

However, all this can be cleared up simply by adopting one further theory: Joe is an incompetent buffoon.

Jerad,

You have repeated the same claim a few times, “Its not controversial, its in Wikipedia” as if that is saying something.

There are so many responses to show why that is baloney, but here is an easy one: Wikipedia says Richard Dawkins knows what he is talking about .

Wikipedia is a propaganda platform for “scientific skeptics” who are never skeptical.

Roy,

You have just made the case for why Cantors idea is BS.

Thank you

phoodoo:You have just made the case for why Cantors idea is BS.It’s not just an idea, but a set of mathematical proofs.

Jared @ 569

Arrogance.

Roy

CharlieM @ 580

I don’t have to know all worlds. If you need a perfect circle, Pi has to be rational, which is not possible so you can’t have a ‘perfect circle in

any world.An approximated circle is what works in the world that you face everyday, that is what is used to engineer your car’s wheels – that is reality.

IMO, there is no ‘ideal world’ where ‘circles are perfect and permanent’. Face the reality.

Jerad:

But then this thread would not go on

ad infinitum…“Within Cantor’s lifetime, logicians demonstrated that set theory is frankly inconsistent. Dangers in mathematics do not get more dangerous than this.”

– David Berlinksi

phoodoo: You have just made the case for why Cantors idea is BS.

Zachriel: It’s not just an idea, but a set of mathematical proofs.

Me_Think:If you need a perfect circle, Pi has to be rational, which is not possible so you can’t have a ‘perfect circle in any world.Not sure why that follows. A circle can be drawn with a string or compass without reference to arithmetic of any sort. It has more to do with lack of perfection in the instrument.

http://momath.org/wp-content/uploads/compass.jpeg

Mung(quoting): “To this day, no one really knows whether this is so. Aproofis unavailable.:Proof of what? What axioms in ZFC do you wish to change?

MT, You have stated that Pi has to be rational but you haven’t said why it has to be so. Why does it have to conform to your idea of perfection which in fact only relates to simplicity? Simplicity and perfection are two different concepts. It would be much simpler if Pi=3 but that would not conform to reality.

In your opinion there is no ideal world, in my opinion there is. You are right that the world of our normal experience is transient but that is because of the way we are constituted and not necessarily because that is the way it truly is.

This thread is populated mostly with comments from cowards, liars, fruitcakes, malignant narcissists and butt kissers. It makes my blood boil. See y’all around. 😀

PS. Sorry KF. I’m pointing fingers at no one in particular. The above is just a general observation. “If the shoe fits” and all that jazz.

An axiom or postulate is a premise or starting point of reasoning.

As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.…

In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction,

nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems.http://en.wikipedia.org/wiki/Axiom

Lets’ agree that we’re talking about faith here, can we?

Zac @ 590

I agree, but they are talking in terms of irrationality of Pi. I am just arguing that since Pi is irrational, there can’t be a ‘perfect circle’ in the sense that circumference will differ – Eg: For radius 2, at 4 digit precision, circumference will be 12.5664, at 20 digit 12.566370614359172954, at 24 12.5663706143591729538506, at 28 digits precision 12.56637061435917295385057353 and so on.

Mung:Lets’ agree that we’re talking about faith here, can we?No. We’re talking about what can be deduced from given axioms. What axioms in ZFC do you reject?

CharlieM @ 592

Please see my comment to zac (@595)

How do you know it is not true ?

Me_Think:I am just arguing that since Pi is irrational, there can’t be a ‘perfect circle’ in the sense that circumference will differ – Eg: For radius 2, at 4 digit precision, circumference will be 12.5664, at 20 digit 12.566370614359172954, at 24 12.5663706143591729538506, at 28 digits precision 12.56637061435917295385057353 and so on.No. We don’t have to measure or calculate pi to make a perfect circle. We don’t have to know if pi is close to three or three and a bit, or even if we don’t have arithmetic at all. If we had a perfect compass and plane, then our circle would be perfect, even if our measurement of the circumference were not.

MT, you are confusing the capability of measuring a certain distance with the actual value of the distance.

Well I can’t say definitely that it isn’t true but to say that what humans experience is a complete view of reality is to say that we are omniscient. You either believe this or you are being inconsistent.

Cantor #581

Thank you for that.

phoodoo #583

Get any introductory set theory book then. I was suggesting Wikipedia because it’s easy and free.

Mung #588

I’ll stick with the majority of mathematicians thanks. It would be nice to have a bit more context for the Dr Berlinski comments reproduced. Since I don’t own any of his books it would be nice if someone could give a bit more of the surrounding text.

Mapou #593

It’s so nice to be appreciated and respected.

I’m used to being told my opinions of evolution are wrong but I am somewhat taken aback that established, non-controversial mathematics can cause such a fuss.

It’s sad to that with the exception of KF, none of the UD regulars have bothered to either support the precepts of set theory or correct their fellows. Oh well, it’s not my blog, I don’t make the rules or police the participants.

From Dr Dembski’s 2005 paper: Specification: the pattern that signifies intelligence, page 5, second paragraph (I’ve added some bold emphasis):

“To see this, consider that a reference class of possibilities ?, for which patterns and events can be defined, usually comes with some additional geometric structure together with a privileged (probability) measure that preserves that geometric structure. In case ? is finite or

countably infinite, this measure is typically just the counting measure (i.e., it counts the number of elements in a given subset of ?; note that normalizing this measure on finite sets yields a uniform probability). In case ? isuncountablebut suitably bounded (i.e., compact), this privileged measure is a uniform probability. In case ? isuncountable and unbounded, this privileged measure becomes a uniform probability when restricted to and normalized with respect to the suitably bounded subsets of ?. Let us refer to this privileged measure as U.”From page 7 of Dr Dembski’s paper:

“Although the combinatorics involved with the multinomial distribution are complicated (hence the common practice of approximating it with

continuous probability distributionslike the chi- square distribution), the reference class of possibilities ?, though large, is finite, and the cardinality of ? (i.e., the number of elements in ?), denoted by |?|, is well-defined (its order of magnitude is around 1033).”Top of page 17:

“This sequence, as well as the totality of such agents, is at most

countably infinite. Moreover, within the known physical universe, which is of finite duration, resolution, and diameter, both the number of agents and number of patterns are finite.”From footnote 12:

“Strictly speaking, P(.|H) can be represented as f.dU only if P(.|H) is absolutely

continuouswith respect to U, i.e., all the subsets of ? that have zero probability with respect to U must also have zero probability with respect to P(.|H) (this is just the Radon-Nikodym Theorem”So I guess Dr Dembski ‘believes’ in countably infinite sets and continuous function.

Mung: Lets’ agree that we’re talking about faith here, can we?

Zachriel: No. We’re talking about what can be deduced from given axioms. What axioms in ZFC do you reject?

Which definitions of axiom do you reject?

Jerad says

It’s sad to that with the exception of KF, none of the UD regulars have bothered to either support the precepts of set theory or correct their fellows.

I say.

Off the top of my head I remember Myself, Mung, ba77, and CharlieM all defending the reality of infinity here and your fellow traveler Me_think actively challenging the concept.

That you did not notice any of this is telling.

peace

Mung,

I think that “faith” is simply the wrong word here. You don’t choose axioms based on whether you believe they are absolute truths or not. It’s more of a question of whether the resulting mathematics is interesting and/or useful.

For example, the somewhat nonintuitive axiom “all triangles have angle sum strictly less that 180 degrees” turns out to be quite productive in certain contexts.

fifthmonarchyman #607

I was talking about the precepts of set theory not just the notion or reality of infinity. And I’m quite sure you misread Me_thinks comments.

But let’s just get to the meat of the matter: do you agree with the notion that you can have a countable infinite set (like the natural numbers), take out a countably infinite set (like the even numbers) and still have a countably infinite set left (the odds in this case)?

Why not just commit on the mathematics and get past the he-said, you-said stuff. Where do you fall on the mathematical fence?

Jerad:

I know and I am pretty sure that is part of the claim we are debating. So just repeating it proves that you are senile.

{1,2,3} a finite set that has the first three positive integers. {1,2,3,…} is an infinite set of positive integers, the pattern which was directly extrapolated from the established finite pattern. There isn’t any difference in mathematics there.

OK how about set subtraction? That seems to be work the same. Is subtraction still part of mathematics?

No, Jerad, the only difference seems to be is that with infinite sets Cantor manufactured a mathy-sounding solution and most people seem to have bought it. Unfortunately it doesn’t do them a world of good but they think it makes them smarter than others cuz they know the troof ’bout infinity stuff by golly.

Please wake us when you find a practical application so we can actually test this concept to see who is right.

Jerad says

Where do you fall on the mathematical fence?

I say,

Cantor’s side. That should be obvious.

In fact I’ve argued in other places that Cantor’s theorem is the reason that you won’t be able to write an algorithm that will fool the observer in my game.

Now I do find the question of whether infinity actually exists to be more interesting than discussions of cardinality. So I don’t spend a lot of time defending Cantor around materialists

Do you think that infinity actually exists or is it just an imagination of the human mind that for some reason just happens to be useful in sophisticated math?

If infinity is just human imagination then discussions of cardinality are “chasing after the wind”.

peace

But you keep repeating your finite version of set mathematics.

Take the natural numbers. Take out all the multiples of 2 starting with 4. That’s taking out an infinite set. But you’ve still got the odds plus 2 and that’s still infinite. Take out all the multiples of 3 starting with . . . 6 is already gone . . . starting with 9. That’s taking out an infinite number of elements but you’ve still got an infinite number of things left. But when I take out elements from a set it can’t stay the same size according to you so it must be getting smaller.

You want there to be an infinite amount of infinities getting smaller and smaller and smaller. With no lower bound. And never crossing over into the finite. And that’s been shown to be unworkable. Go read an introductory book on set theory. I shouldn’t have to type a whole textbook into this blog just for you.

Mathematics with infinite sets works differently. And you can easily learn about this by LOOKING IT UP.

Yes but it works differently with infinite sets.

There were so many concepts and ideas and things which challenged me and really made me think when I took mathematics courses. It wasn’t spiritual, it wasn’t philosophical it was . . . trying everything out and seeing where it went.

Uh huh. Joe, why can’t you support your view of set theory mathematics with some references? Why haven’t you addressed the theorem I linked to? Why do you use terms in non-standard ways without giving some examples, at least, of why you’re justified in doing so.

If your life is only going to be defined by what is practical then it’s going to be very flat and dull in my mind. Can you point to a practical application that comes out of believing in a cosmic designer? Can you show me some practical reason for reading Shakespeare or Dante or Melville or Poe or Hawthorne? Can you give me a practical reason for viewing the paintings of Raphael, Titian, Caravaggio, Vigee Lebrun, Pollack, Magritte? Why should we watch films like 2001 or The Day the Earth Stood Still or Syriana? Why should we care about black holes or CMEs or exo-planets or comets? None of this stuff affects us on a day=to-day basis. Let’s just throw it all out and spend our time and money on practical things.

To paraphrase KF again: God is a kick-ass mathematician. As you would expect.

fifthmonarchyman #611

I would very much like to hear the reasoning behind your alias.

I was just checking.

I’m not sure that follows but . . . okay.

What does materialism have to do with it? We’re talking pure mathematics!!

Honestly, I don’t understand why this is coming up at all.

What is sophisticated math? Do you mean anything over a high school level? ‘Cause there’s a lot of math past that.

I think that we cannot experience infinity as a physical reality. But IF you’re going to do mathematics then you damn well better comes to terms with it, knock it about a bit and wrestle it to the ground. Because otherwise you end up just standing at the edge of a cliff (up or down?) not moving.

I think any intelligent creature who starts down the mathematics path is going to come up against these same issues. I think the questions are universal.

What’s absolutely incredibly amazing is that, within certain parameters and given certain axioms, mathematicians have come up with ways of thinking about infinities that are consistent and useable. Not useable as we experience driving to work or playing with the kids or paying bills. But useable in the exploration of the mathematical landscape.

It’s a great place to wander around.

Jerad says

Honestly, I don’t understand why this is coming up at all.

I say,

If infinity is not real you have just spent over 600 comments having a “how many angels on the head of a pin” discussion

if infinity is just a synaptic buzz in the human brain there is literally no way to demonstrate Mapou and Joe are incorrect.

If you want to prove them and those like them wrong you need to demonstrate that things outside the cave have a real existence.

I just don’t think you can do that given your presuppositions

You say,

What’s absolutely incredibly amazing is that, within certain parameters and given certain axioms, mathematicians have come up with ways of thinking about infinities that are consistent and useable.

I say,

You do know that my worldview has an explanation for that phenomena don’t you?

peace

Jerad says

To paraphrase KF again: God is a kick-ass mathematician. As you would expect.

I say

Exactly

quotes:

That God is of himself, that is, neither from another, nor of another, nor by another, nor for another But is a Spirit, who as his being is of himself, so he gives being, moving, and preservation to all other things, being in himself eternal, most holy, every way …….infinite…….. in greatness, wisdom, power, justice, goodness, truth, etc.

1644 LBCF

and

“I was merely thinking God’s thoughts after him.”

Johannes Kepler

end quotes:

peace

Zachriel:What axioms in ZFC do you reject?Mung:Which definitions of axiom do you reject?Not an answer. The definition in inherent in the question. Axioms are postulates. What axioms in ZFC do you reject?

fifthmonarchyman:if infinity is just a synaptic buzz in the human brain there is literally no way to demonstrate Mapou and Joe are incorrect.Cantor already demonstrated they are wrong.

Zac said,

Cantor already demonstrated they are wrong.

I say,

To who? Apparently not to them

peace

.

Ditto that.

.

Aurelio Smith,

Jerad quote: “I’ll stick with the majority of mathematicians thanks.”

This is not erudition, He is simply displaying typical ‘scientific skeptic” cult behavior. They pretty much run science departments in this country if you don’t know.

One of their propensities is to get together and say, “Ok, these are the things we need to believe in to maintain our materialist worldview, which is critical. Under NO circumstances are they to be questioned, because then we will have to be forced to ponder the world we live in. That is unacceptable! Neil Degrasse Tyson, Jerry Coyne, Seth Shostak, Steven Novella, Richard Dawkins, Danniel Dennet, Evan Bernstein, Bill Nye, Penn Jillette, Ira Flatow, Lawrence Krauss, Jad Abumrad, get out there and spread the word. Here is the list of the things we believe in…..

Be unified. It is imperative that no one waivers even a fraction, regardless of any so called evidence. We must be strong! Oh, and call yourself a skeptic for crying out loud, the irony helps to confuse people!!”

Apparently no member of the set {Joe, Mapou, phoodoo} ever pauses to think “If I’m in a set with these two bozos, I must be doing something wrong.”

Keiths- (A member of the erudite “scientific skeptic” cult.)

“I, I , I need to stop this discussion now! It’s in direct confrontation with my worldview of never questioning materialism, or the founder of our great movement, the plagiarist Darwin, and his modern day disciples. But, but, I have nothing to say, I don’t even really know what mathematics is…what can I do?? Oh well, I will just fall back on my usual strategy, just bang my head on the keyboard and hope it strikes some keys. It always works for me in the past. I don’t see why I always get banned everywhere, I can’t help what my forehead hits.”

CharlieM @ 599

I have no idea what you are referring to.

They have to read and understand Cantor’s proof. Cantor’s ghost can’t come and explain personally to them.

fifthmonarchyman#614Well Dr Dembski clearly understands and uses countably infinite sets and he teaches divinity now I believe.

Also I cannot force someone to explore and question their own beliefs. I cannot summarise whole textbooks of materials on a blog. I have tried to explain and answer questions to the best of my abilities.

Aurelio Smith#617Thank you. I keep thinking of things I could have said better! Ah well.

Aurelio is such a lovely name.

Cantor#619Thank you. Your alias is very comforting at least!!

phoodoo#620Since you have already made up your mind I shan’t bother to try and discuss these issues with you anymore. But I would strongly suggest you get an introductory book on set theory and read it with as open a mind as possible. It is a beautiful and mind stretching field of mathematics. Personally I prefer number theory but when in Rome . . .

phoodoo#622You have a very strange view of some of your fellow human beings.

MT, I obviously didn’t make myself clear @ 599.

You wrote @ 595:

In the above statement you are saying that a circle will expand or contract depending on how accurate Pi is expressed. The circumference will not differ. The ratio of the radius to the circumference of a circle is fixed for eternity, it will never change. What has changed above is your precision in expressing Pi. Pi is not the variable, you are.

CharlieM @ 626

The values given are the circumference of the circle. Circumference will differ as Circumference is 2*Pi*r. Since circumference differs, ratio of radius to circumference too will differ (hence it can’t be a ‘perfect circle’).

I think it would be better if you define what you mean by a ‘Perfect Circle’.

P.S: If you mean Pi’s precision is fixed before hand, then of course, circumference and ratio will not change, but what is the ‘perfect precision’ for the circle ? Is there a limit to the precision which will satisfy the definition of perfect circle – assuming you have one?

MT, You have it backwards. Pi does not determine the form of the circle, it is derived from the form. Pi is derived from the circumference divided by the diameter. This value will never change regardless of our ability to measure it precisely.

Why can it not be measured precisely?

Well if we could somehow obtain a piece of thread with zero thickness that was exactly the same length as the diameter and we laid it exactly on the circumference, we could measure three lengths but there would still be part of the circumference left over. So we divide our thread into ten equal pieces. We could then try to bridge the gap left but there would still some left over. So we further divide the thread we are using into ten equal pieces. We find that four pieces will almost bridge the gap but we still haven’t reached our starting point. No matter how many times we divided our thread in this way we would never reduce the gap to zero.

In a perfect circle the distance from the centre to the circumference with always remain the same no matter what point of the circumference the radius touched.

LoL! @ Jerad- Dembski did NOT use the concept tat all countable and infinite sets have the same cardinality. THAT is what we have been debating.

Also, Jerad, by your “logic” Cantor was wrong because he couldn’t reference anyone already using his concept!

Jerad sed that the mathematics of infinite sets is different yet I have provided examples that refute his claim. So what does he do? Ignore it and prattle on.

fifthmonarcyman:To who? Apparently not to themYes, to them. That’s the nature of mathematical proof.

I know and I am pretty sure that is part of the claim we are debating. So just repeating it proves that you are senile.

Jerad:

I don’t have a finite version of set mathematics. However I will repeat my refutation of your concept every time you repeat that concept. It only takes ONE refutation, meaning I can repeat the one that works until you face it.

I know and I am pretty sure that is part of the claim we are debating. So just repeating it proves that you are senile.

Jerad:

I don’t have a finite version of set mathematics. However I will repeat my refutation of your concept every time you repeat that concept. It only takes ONE refutation, meaning I can repeat the one that works until you face it.

How many natural numbers are there?

There are more natural numbers than there are natural even numbers. It is all relative

Consider a hotel with

~~an infinite number of rooms~~a room for each of the natural numbers, and all the rooms are filled. A new guest arrives. We can move the guest in room 1 to room 2, the guest in room 2 to room 3, the guest in room N to room N+1, and so on. Now room 1 is open for the new guest. We can do this again and again. The Grand Hotel is always full, yet always has room for more.A new natural number is born! I told you this was magical stuff!

Folks, passing by briefly. As already shown, the naturals, evens, odd (and more) are countable and for each there is no last such number, there is always one more, then another endlessly. That yields a common cardinality, aleph-null, which takes the just outlined meaning. To try to extend to such a situation how we scale finite countable sets will fail. Beyond, we can define further sets such as the set of subsets of naturals, which will have scale 2^aleph-null. This may plausibly be seen as the continuum number as it takes in all ordered or structured n-tuples [just suitably assign meanings to the list of any given subset per decimal place value assignments and the requisites of co-ordinate systems X,Y, r-theta, x1, jx2, etc] such as would specify co-ordinates of points in the plane or 3-d space or any ball in a vector space of desired dimension] etc. Of course that is not proved. KF

Hehe.

It would seem to me that…

Nonsense + anything else = nonsense

Physicists hate infinity. They do their best to “cancel infinities out.” Which tells me what it’s really all about is symbolic manipulation where “infinities” are merely placeholders for nonsense which seems to conform to

something, but nobody know what “it” is, and stick a label of “infinity” to it.Magic, I tell ya.

But what do I know.

Zachriel’s hotel must be part of the Magic Kingdom…

“True” in “theoretical concept”, false in actualization. The problem is, nothing can

actuallycount them. There will always be a disconnect between what is “countable” and a count actually getting accomplished in the real world.The nonsense comes into play (and why physicists hate infinity) is when you try to actually get real-world

meaningout of an “infinite but countable” set. Matters are “worse” with the aleph one set, which is “non-countable”, i.e, no mapping from aleph null to aleph one.mike1962:Nonsense + anything else = nonsenseYou didn’t answer the question. How many natural numbers are there? Is it finite? Or not finite?

The question wasn’t directed to me, but I will say

intuitivelyit seems the answer is “infinite”, but I don’t really know what that means beyond what seems like an algorithmic definition that has no actualization in reality, and which physical scientists make serious attempts to avoid and “cancel out.”So my answer: Hell if I know.

mike1962,

Physicists also have to pay close attention to the distinction between the countably and uncountably infinite. Hilbert spaces with countable bases are ubiquitous in QM; Hilbert spaces with uncountable bases are generally avoided for reasons which are mostly beyond my grasp, but my understanding is that the uncountable case introduces some very difficult problems that are best avoided.

mike1962:The question wasn’t directed to me, but I will say intuitively it seems the answer is “infinite”,Okay.

mike1962:but I don’t really know what that means beyond what seems like an algorithmic definition that has no actualization in reality,In mathematics, it’s called induction. For each natural number n, there is a natural number n+1. This is also called the axiom of infinity.

mike1962:and which physical scientists make serious attempts to avoid and “cancel out.”Physicists have often used infinities, such as for solving integrations. See Newton, Mathematical Principles of Natural Philosophy, Philosophical Transactions of the Royal Society 1687.

Zac before

What Paradoxes?

Zac now

The Grand Hotel is always full, yet always has room for more.

Me now,

oh the humanity 😉

peace

Joe: There are more natural numbers than there are natural even numbers. It is all relative

Joe, there are just as many even numbers as natural numbers.

You’re basically restating ground already covered. But thanks.

Sure, they have gotten “used”, and still do, but they are not allowed to remain in any practical results, since that is nonsense. I’m not saying “infinities” don’t exist in some sense, and are symbolically manipulated, but what I think they really are beyond symbolic and algorithmic definition and the intuition involved in our minds with respect to them, well, I think I already covered that.

…. that is to say, no instantiations of infinite sets.

M62: Prob is of course, once calculus came in the door so did all of this implicitly. And sans calculus and its extensions, bye bye modern world. I suggest the best stance is to recognise that mathematics addresses an ideal space of forms, with a logical structure, indeed Mathematics can be defined as the logical and sometimes quantifiable study of structure. Then, we can extend and apply mathematical models to the real world, recognising that Plato et al had a point (though not the whole story) when they spoke of a world that imperfectly mirrors the ideal abstract world. Or in terms that are more relevant to us, mathematics is about a common, shared mental-logical space which has powerful relevance to the experienced world of our common physical existence; e.g. that a square circle is logically impossible and therefore physically unrealisable even to approximation (e.g. by bending a paper-clip), is a seemingly simple case with all sorts of subtle but powerful implications that come out on pondering what it points to. Which has in it all sorts of suggestions as to the roots of our world and how it comes to be so ordered. From my view the first thing, almost, that God is, is a Mathematician. KF

PS: I think also that the point on what cardinality of a set is, is that it speaks to scaling of a set, leading to the point that two sets share common cardinality if they may be exactly matched in their members in some orderly way. For sets that are exhaustible in finite steps, that leads to you can count the sets. But when we have sets that are transfinite, we can only set up a set-builder notation or assignment that shows the logical status of that correspondence or otherwise. Once that is done, we can decide whether or no a given thing is in the set and we may assign a match to a standard set such as the natural numbers. It is then a logical demonstration that due to the inexhaustibility, the naturals match the evens and odds, etc, also the rationals. But, once we go over to continuum, we have ways to show that we can exhaust the naturals but will always have further members left over, i.e. the real numbers are in a continuum and have a higher order cardinality. I have already suggested that by looking at the set of subsets of the naturals, we may get a glimpse of how an ordered structure and functional interpretation on such subsets (which BTW will require repetition of digits) will allow us to in principle address every point in a space assignable a coordinate system. So, the continuum can reasonably be seen as arguably having cardinality c = aleph_1. But that is light years away from an actual proof.

I’ve been reluctant to continue commenting on this thread given the dishonesty and gutlessness of some of the commenters but this comment by Zachriel is interesting because it reveals the crux of the problem with infinity.

Zachriel:

Of course, this does not prove that anybody can build an infinite set or that an infinite set exists anywhere. It’s just a rule that says you can add 1 to any natural number n to obtain a new natural number that is neither n nor 1. The rule is self contradictory for the following reason:

To any natural number n, one can also add ANY natural number such as 2 or 3 or 99 or 2422435, etc. The only exception is infinity. Why? It’s because adding infinity to n does not yield another natural number. It yields, well, infinity.

In other words, if infinity is a natural number that exists, one cannot add a natural number to it to yield a new natural number. The so-called

axiom of infinityis broken before it even gets out of the gate. 😀Mapou,

Infinity is not a natural number.

Mung, the reason for the disconnect is that Joe is using his own private definition of “more”. He is developing his own infinite set theory. (e.g. See Jared’s post 658 below)

Mapou:if infinity is a natural number that exists one cannot add a natural number to it to yield a new natural number.Infinity is not a natural number: It’s the cardinality of the natural numbers. So, no. It’s not broken.

Of course. This is why it does not exist.

Z, pardon, the cardinality of set N is aleph-null, not something so broad as “infinity.” The issue here is to address inexhaustible sets and do so reasonably in a way that does not toss Calculus. That will lead to what Cantor et al achieved or something very much like it. KF

Mapou:This is why it does not exist.How many natural numbers are there? Is it finite? Or is it not finite?

Joe,My reading of your interpretation of set theory is that the cardinality of the natural numbers is bigger than the cardinality of the evens which in turn is bigger than the cardinality of the multiples of 4 which is larger than the cardinality of the multiples of 8, ad infinitum?

Is that correct? All the sets listed above are infinite but their cardinalities are decreasing?

If I continued the process (multiples of 16, then 32, then 64 . . . ) I’d keep getting smaller and smaller cardinalities but I’d still have infinite sets.

Correct?

Is there then a lower bound to the cardinalities or do they continue to decrease without ‘hitting a wall’ so to speak. If there is a lower bound do the cardinalities hit it or do they approach it asymptotically?

I can’t answer this question because this is your version of infinite set theory. Only you can answer it.

kairosfocus:the cardinality of set N is aleph-null, not something so broad as “infinity.”Aleph numbers are measures of the cardinality of infinite sets. Natural numbers are one such infinite set.

c=aleph_1 (Continuum Hypothesis) has been shown to be

undecidablein ZFC set theory.Zachriel (completely and dishonestly ignoring my point that, if a size n is not a natural number, it does not exist) retorts:

The question is self-contradictory because it assumes that which it is trying to prove. It assumes that natural numbers can be fully counted to determine a quantity or size. It’s a dumb question.

Mapou:It assumes that natural numbers can be fully counted to determine a quantity or size. It’s a dumb question.So you’re saying asking if the natural numbers are finite is incoherent? Can we assume that if there is an n, then there is an n+1?

Mapou,

Eh? Does the number 1/2 “exist”? It’s not a natural number.

daveS, you may have a point, although I think it’s a tenuous one. To remove any ambiguity, I corrected my argument. I was thinking of a number used to express a size or quantity. It cannot be 1/2 or -1 or any other non-natural number.

Zachriel:

No. I’m saying that the idea or claim that there is a set of all natural numbers is incoherent. Your question assumes that an infinite set already exists, the very thing you are trying to prove. It’s a dumb question.

Mapou:I’m saying that the idea or claim that there is a set of natural numbers is incoherent.So, there’s no set of rational numbers or real numbers, or prime numbers?

Zachriel:

Only finite sets can exist. There is no set of ALL natural numbers (Note: I corrected my comment @665 to say “all natural numbers”). If you had an infinite set of ANYTHING, you would be able to show it to me. You can’t. Saying that you can imagine it in your head is not proof of existence.

Mapou:Your question assumes that an infinite set already exists, the very thing you are trying to prove. It’s a dumb question.In set theory, the assumption is that if there is a n element, there can be an n+1 element. This implies an unbounded set. If you reject this, then you reject much of mathematics which has relied on induction since the Classical Greeks.

Mapou:There is no set of ALL natural numbers (Note: I corrected my comment @665 to say “all natural numbers”).You can have set theory with or without the axiom of infinity. When talking about set theory, it normally includes this axiom, and there is no inconsistency in adopting this axiom, even if you don’t like it.

Zachriel@ 666 ( an omninous post# 😀 )The answer, of course, is sure there “is”, if you create a definition for the set. It comes down to coherency and contradiction with respect to pure mathematical definition, an actuality when it comes to the real world we live in.

The former involves Cantor and Godel

et aland is admittedly very interesting fodder for mental masturbation. (I would put the discussion of the nature of such squarely in the realm of philosophy, which many think is “dead.” Apparently not. Although there does seem to be real world implications if we just take it as tricky symbol manipulation and not worry about “what it really means.”)The latter pertains to instantiation of any infinite set which seems to me to be impossible nonsense in the real world which I think boils down to a tricky manipulation of symbols.

Intuitively, there seems to be some sort of “Platonic reality” to infinity (of the various kinds) and numbers in general. Sir Penrose thinks our intuition is touching upon something transcendent and real when it comes to the subject of numbers and has spilt a lot of ink saying why he thinks so. Maybe he’s right.

But what do I know.

Zachriel:

I don’t reject the concept of an expanding set at all. I have no problem with continually adding a new member to a set. But regardless of how many new members you add, the set is always finite.

Zachriel:

I really have no problem with the axiom. I have a problem with the use of the word “infinity” in its name. It’s a misleading and self-contradictory name, IMO. In fact, I may even agree with most of Cantor’s ideas on expanding sets. I just disagree that he was working with infinite sets or that it is possible to work with infinite sets.

Mapou,

I don’t reject the concept of an expanding set at all. I have no problem with continually adding a new member to a set.

In your view, is it legitimate to regard 72,944,303,471,110 as a natural number? After all, you’ve never seen anyone construct that number, starting from 1, by repeatedly adding 1.

keith, sorry but I don’t see what your question @673 has to do with infinity or infinite sets. If you need a definition for natural numbers, Google is your friend.

Zachriel:So, there’s no set of rational numbers or real numbers, or prime numbers?mike1962:The answer, of course, is sure there “is”, if you create a definition for the set.That’s what ZFC set theory does with the axiom schema of

~~replacement~~specification.mike1962:The former involves Cantor and Godel et al and is admittedly very interesting fodder for mental masturbation.As pointed out above, infinity is important in physics. See Newton, Mathematical Principles of Natural Philosophy, Philosophical Transactions of the Royal Society 1687.

Mapou:But regardless of how many new members you add, the set is always finite.The set of natural numbers is unbounded.

Mapou:I really have no problem with the axiom. I have a problem with the use of the word “infinity” in its name.Seriously. The name?

If I might dive in, personally I think 72,944,303,471,110 is natural number. I can write a computer program to count up to it and display it on my KayPro (okay, maybe I would need a faster computer than that to do it), which is proof enough for me that it is a value with meaning in the real world. I cannot do that with all (or even “most” of) the values in a transfinite set.

On a slightly more serious note, 72,944,303,471,110 is obviously natural because it’s quantified and obviously so. No reason to even count up to it. It’s already there! 🙂

Zachriel:

It is still finite regardless of how you call it.

What the name implies.

Mapou:It is still finite regardless of how you call it.Before you said it was incoherent. Now you say it is finite. Yet it’s obviously unbounded.

Mapou:What the name implies.In mathematics, it means unbounded or without limit.

Cantor, yes. I am simply pointing out a plausible way of looking at the matter for those whose approach is modelling and empirically oriented. KF

Oh, I most definitely agree, and already acknowledged that. There is symbolic manipulation thereof that has led to useful, real world results in physics. However, the results themselves do not contain infinities. Infinities must be cancelled out. They must be tossed out on their ears by the time any usefulness of the exercise is seen, either in terms of our understanding of the world, or our ability to make better toasters.

It is the nature of what is actually being manipulated which is the controversial thing in my thinking. Are infinities that are represented by the symbols mere mathematical tricks? Or do they correspond to something that our intuition is picking up on that is profoundly (and by necessity, if true, transcendentally) real? As I said, Sir Penrose thinks the latter and has spilt a lot of ink explaining why he thinks so.

My best friend has a PhD from Harvard University math department with a specialty in topology. (Weird stuff. And mostly useless with respect to practical reality. However, he knows his maths.) We have spent quite a number of hours discussing infinity, and I think I understand the things he tries to explain to me. I’ve also read a lot of books that deal with the subject. In the end, for me…

I think it’s

practicalnonsense, symbolic trickery, with perhaps some sort of correspondence to something else that may be real that transcends the physical world. After all, 1 + 1 = 2 seems to transcend any physical instantiation, so maybe transfinite sets do too. But I have yet to see an instantiation of a transfinite set. Nor do I see how it could even be possible in theory.But what the hell do I know.

Mapou,

It’s simple.

You write:

If you accept the existence of large and physically unrealizable finite quantities such as a googolplex, what is your principled reason for rejecting infinity?

Zachriel @678, I don’t care how you call a set or series that can expand. It is finite. I’m getting tired of your vapid replies that add nothing to your arguments for the existence of infinite sets or infinity, arguments that I have already refuted.

keith:

All finite quantities can be written down (or represented) in a sufficiently large finite universe. Infinity cannot be written down in any finite universe. Any number yields another number if you as 1 to it. That is, any number except infinity. Why is infinity the exception? Answer: it does not exist. Live with it.

Now, I may be stating the obvious, but it seems to me that an instantiation of a transfinite set as an

actual ontologywould necessarily require trans-temporalityinan unbounded reality orasan unbounded reality.Platonism anyone?

Mapou:

If the universe is finite, then some finite quantities are too large to be represented within it. Do those quantities exist, according to you?

If so, then why not infinity?

mike1962:Infinities must be cancelled out. They must be tossed out on their ears by the time any usefulness of the exercise is seen, either in terms of our understanding of the world, or our ability to make better toasters.No, they’re not tossed out on their ears, but essential steps to finding the answers.

mike1962:Are infinities that are represented by the symbols mere mathematical tricks? Or do they correspond to something that our intuition is picking up on that is profoundly (and by necessity, if true, transcendentally) real?In the calculus, they represent dividing the continuum into smaller and smaller pieces, to the limit.

mike1962:After all, 1 + 1 = 2 seems to transcend any physical instantiation, so maybe transfinite sets do too.Yes, mathematics is an abstraction, and is a model when it conforms in one way or another to the world. But all models are incomplete and imperfect. Just because zero causes some people philosophical discomfort doesn’t change this. People get use to it, like the Earth moving, or negative numbers.

Mapou:I don’t care how you call a set or series that can expand.Unbounded sets are normally called infinite.

Mapou:arguments that I have already refuted.Actually, your argument was that you didn’t like using the term “infinity”. There are many problems in mathematics and physics that require the use of unbounded limits, by whatever name you call them.

Having re-read my words, I think I was not clear. They have to be cancelled out by the time the equations have any

finalvalue. The time-worn issue of integrating General Relativity and Quantum Physics is an example.But that’s just restating the “problem” using different words. Symbols representing “the continuum” are manipulated. But at the conclusion, the infinities are out of the picture.

I never said the manipulation of symbols that respresent infinities were not useful. Of course they are, else nobody would be bothering to discuss them here.

I would have to agree.

I don’t think I would put those at the same level of abstraction. Neither of those require trans-finite space and trans-temporality to conceivably have an instantiation. Some ideas are more abstract than others. Infinities are the ultimate abstractions.

keith:

No. Nothing exists unless it does. It’s that simple. If you can show me your infinite set (or any set, for that matter), it exists. Otherwise, it’s a figment of your imagination. Adding ‘…’ or ‘etc.’ at the end of your series does not count.

Zachriel:

I don’t care.

You are lying.

I still don’t care.

mike1962:I don’t think I would put those at the same level of abstraction. Neither of those require trans-finite space and trans-temporality to conceivably have an instantiation.But that’s just restating the “problem” using different words. Symbols representing “negative sheep” are manipulated. But at the conclusion, there’s no negative sheep.

Mapou:I don’t care.Before you said, “I don’t care how you call a set or series that can expand.”

Zachriel:Actually, your argument was that you didn’t like using the term “infinity”.Mapou: “I really have no problem with the axiom. I have a problem with the use of the word ‘infinity’ in its name.”

Mapou:I still don’t care.Good. Then we will continue to call it “infinity” consistent with standard usage.

Are you saying you don’t think there’s a difference in the level of abstraction with regards to infinities and negative numbers?

mike1962:Are you saying you don’t think there’s a difference in the level of abstraction with regards to infinities and negative numbers?If by abstraction, you mean there are no negative sheep, then yes. The reaction is analogous to when negative numbers were introduced.

No, I was referring to the

levelof abstraction (you do think there are levels of abstraction, don’t you?), and whatyouthink about it, not reactions from some dude in the past.Wanna try a different answer?

Zachriel:

I still don’t give a rat’s asteroid. No matter what you call it, it is still finite.

Mapou,

If you prefer to work in this very austere mathematical environment, that’s fine, of course.

You stated earlier that physicists always end up using discrete methods, despite the fact that they sometimes claim otherwise. Do you think it matters whether physicists specifically accept the “existence” of infinity? In the end, will there be any difference?

Mapou #688,

Okay. Then by your logic, there must be a largest finite number. What is it, and what happens if you add 1 to it?

mike1962:No, I was referring to the level of abstraction (you do think there are levels of abstraction, don’t you?), and what you think about it, not reactions from some dude in the past.Abstractions are abstractions. Negative sheep don’t exist. Zero sheep don’t exist.

mike1962:You stated earlier that physicists always end up using discrete methods, despite the fact that they sometimes claim otherwise.Physicists often work with infinities. See Newton 1687.

mike1962:what you think about itWe’re rather fond of the transcendental numbers.

daveS:

What is austere about it? I use the same calculus that everybody else uses. I see no infinity in it though.

No, of course. Physicists can claim anything they want. In the end they die like everybody else but the truth remains.

keith:

Any number that you add 1 to becomes a new number. You can’t do that to infinity because it does not exist. If you can represent a number in some manner, the representation exists. I don’t know what the largest number that has ever been represented is but, you know what? I don’t really care. Your point is what again?

daveS:

I just now realized that you asked me this question before but in different words. I should say that it makes no difference as far as the non-existence of infinity is concerned. But it makes a HUGE difference in the way we understand and look at reality once we realize that reality is discrete and finite. I already answered this question @255 above.

Mapou #700

You’ve never taken an integral from something to infinity?

Taylor series are sums from one to infinity. Einstein used a Taylor series in his derivations.

The area under a normal curve is one if you integrate from minus infinity to infinity.

Gravitational and electrical and magnetic fields extend out to infinity. There is nothing in their formulas which says they end some place.

I’m finding this conversation a bit weird. Limits, asymptotic behaviour, you can’t work with these things unless you can deal with the infinity large and the infinitely small. How do you know if an infinite series converges or diverges? The famous one, Zeno’s paradox, is a classic example:

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + . . . = 1

How do you establish that? You give me an tolerance, a distance from the answer 1 that you want me to get to and then I tell you how many terms of the sequence I need to add up to get to that tolerance. AND I can do that no matter how small you make that tolerance.

The basic concept of the limit underlies ALL of calculus since a derivative is defined by a limit.

Sigh.

Never mind. A lot of good work is and can be done just by utilising techniques developed by the theorists. I know that. But the theories have to be solid and many of them sit on the infinitely large and the infinitely small. And, guess what, it works.

I agree that negative sheep don’t exist. But I do think that zero sheep can exist. Class of “sheep.” Instantiation of class of “sheep.” Zero instantiation of class of “sheep.” But I cannot imagine a negative instantiation of the class “sheep”, can you?

Anyway, do you think these are on the same level of abstraction as infinities, and infinities of infinities? If so, we’ll have to simply disagree.

No, I didn’t. Not at all.

Agreed. (For the third time.)

I don’t think you’re carefully reading what I write.

Me too.

Are you a platonist with respect to numbers? (Lowercase P.)

Jerad @703,

You can jump up and down and foam at the mouth as much as you want but I see no infinity in your argument. Nothing ever reaches infinity. Keep jumping.

However, these “infinities” that theories “sit on” are derivations of definitions of transfinite sets. Symbolic definitions. Nobody ever

actually dealswith anyinstantiatedinfinities. No matter how you slice it, it boils down to manipulation of symbols.They are amazing. And they work. But why do they work?

The “controversy”, for me, and some others, is whether or not infinities have a real, you might say, “platonic” reality or ontology, regardless of the symbolic definitions humans have devised.

Stuff like that.

I believe that Mapou is objecting to an instantiation of any infinite class or set in the Real World that we inhabit, which, of course, I think he would be correct, if the Real World is limited, and most physicists think it is.

Stuff like that

But what do I know

All in good fun

Mapou#706I know you disagree with me but it would be nice if you were a little less rude. Foaming at the mouth? I know moderation is practically non-existent on UD at the moment but . . . please.

You have seen and done definite integrals out to infinity though? You have had to deal with the concept. You must have done. Did you just learn what hoops to jump through and bite your tongue for years of higher education?

mike1962#707Well, how did Cantor show that the size of the set of real numbers is larger than the set of integers if he didn’t deal with infinities?

What about the hyperreal numbers?

And the surreal numbers?

And the imaginary numbers?

The irrational numbers have infinite, non-repeating decimal expansions. The Greeks figured out something was weird about sqrt(2). Same with Pi. And e.

Have you read about Cantor’s continuum hypothesis?

Because they are sitting on a firm theoretical foundation which transcends the material world?

Well, using them works and gives good, solid, practical, useable results. And we’ve now got a solid underpinning for how to work with them.

But surely he would agree that’s there’s more to existence than the physical, ‘real’ world. He sounds like the real materialist!! Seriously, I thought those with a more theological outlook than myself would embrace and welcome discussions of the infinite. But I do find this whole conversation confusing regarding what some people are saying. I should probably just shut up.

I tell what I really don’t get is the hostility to some of the ideas I’ve been trying to explain.

Okay, maybe you don’t understand what I’m talking about or like it or care or think it’s important. But why the hostility from people like Mapou and phoodoo? Why the association with materialism?

(In fact, with his insistence on solid, real world, measurable quantities Mapou sounds like the real materialist.)

If you really want to have a dialogue and you want to discuss ideas but you treat some with scorn, derision and outright hostility then you can’t really expect people you disagree with to want to talk to you. I’m beginning to wonder why I bothered. ‘Foaming at the mouth’ Sheesh.

Jerad:

I refuse to be polite to you, Jerad, because I have seen your dishonesty in this thread. It is insulting and offensive to me.

No. I, too, used to believe in the nonsense.

mike1962:I agree that negative sheep don’t exist.Okay.

mike1962:But I do think that zero sheep can exist.Zero sheep is indistinguishable from zero marbles.

mike1962:Anyway, do you think these are on the same level of abstraction as infinities, and infinities of infinities?The way abstraction has been used on this thread is to refer to mental constructs. Negative numbers, zero, and infinity are mental constructs, that is, abstractions with no material reality.

As we pointed out, people once had troubles with negative numbers, then decided to use them but pretend they were just computational devices, until today, they seem just as normal as natural numbers.

mike1962:No, I didn’t. Not at all.Sorry, misattribution.

mike1962:Are you a platonist with respect to numbers?If wishes were fishes, the oceans would be riches.

mike1962:No matter how you slice it, it boils down to manipulation of symbols.Sort of like long division.

Mapou,

Well, with no infinite sets, it seems to me your system would lack many of the standard tools of mathematics. For example, the Hilbert spaces I referred to above, whether separable or not, make no sense if infinite sets do not exist. They are complete inner product spaces over R or C by definition, so the underlying set of vectors must be uncountable. As a specific example, l^2 (little L 2).

Maybe you can kludge together some sort of discrete approximation to things such as these, but it seems like a lot of work if in the end, you get approximately the same answer.

Are there _any_ existing theories in physics that _don’t_ involve so-called “infinite” sets? Like I said, I know nothing about physics, so maybe there are; if so, I’d like to know about them.

Zachriel @ 711

I like that post. Esp about the fishes and riches.

Zero is not even a number. Therefore zero is not even. QED.

Mapou,

I was also wondering how you would deal with sequences. For example, the sequence of partial sums of the harmonic series. How would you prove that it diverges? Or do you believe that it diverges in the first place?

The only rational explanation seems to be that he has not yet made the transition to adulthood. His illogical and insulting posts do not seem to be the product of a socially mature adult mind. This behavior is not confined to this thread; it pervades all his posts here at UD.

You are wasting your time trying to have a logical and civil dialog with him. He is neither logical nor civil.

Let’s try this: have a dialog with me instead, and totally ignore anything that Mapou posts. Let’s see where it goes.

.

Actually, you managed to quote my post before I edited it. But I cannot say that I object to anything you said in your reply.

Nawwww. What fun would that be?

Not sure why Mapou gets worked up about the subject.

I am admittedly quite disposed to the platonic (small P) side of the fence with respect to all of this. I would have to say I agree with Mapou in certain ways with regards to instantiation of transfinite classes and sets. But my intuition tells me the classes and sets have an platonic ontological reality beyond mere symbol crunching, as non-commonsensical as it may seem to common sense.

I heard that Cantor claimed that God revealed his transfinite ideas to him. Does anyone have a source for this?

All in good fun.

All in good fun.

daveS @715, I have Mathematica on my computer and I can use it to do all sorts of beautiful things that supposedly assume the existence of infinity. And yet, I can assure you that my computer is as discrete and finite as can be. Like I said earlier, I use Fourier analysis to decompose waveform data into a frequency spectrum. Guess what? I feed it discrete values and I get discrete values in return. No infinity and no infinitesimals ever enter the picture.

How is that possible? The reason is that it is a lie that infinity is ever used in any of the calculations. And it is not just a small lie or a simple misunderstanding. It is a big lie, a HUGE LIE.

Poor child, never had to grow up.

Mapou:And yet, I can assure you that my computer is as discrete and finite as can be.Sure. That’s what we mean by a set, treating a collection as a single entity. So the set of mammals, the set of cat’s eye marbles, the set of natural numbers. We put them between curly brackets and then discuss them as a single entity.

It’s in the nature of humans to divvy up the universe. Set theory is how we add mathematical rigor to these divisions.

Zachriel @720:

I was waiting for something of substance. Silly me. How does that prove that infinity is required to do ANY kind of math when I just proved the opposite?

Daniel King:

Nah. It would be childish if I started whining to the moderator or to your momma.

Mapou,

It’s been a while since I used Mathematica, but I agree it’s an amazing piece of software.

Can you give some specific examples of these calculations that supposedly assume the existence of infinity?

I do understand that you can compute Fourier coefficients (approximately at least and sometimes exactly) without this assumption.

(edited to make a correction)

Mapou:How does that prove that infinity is required to do ANY kind of math when I just proved the opposite?We simply responded to your suggestion that computers can’t work with the mathematics of infinity. They can — the same way people do — by defining a set to contain a non-finite assemblage, such as the natural numbers.

As for most computer integration, that’s usually done numerically which provides for approximate answers. However, it’s possible to solve some integrations using the calculus of the infinitesimal, something Newton used to great effect in Principia.

Cantor:

I don’t care whether you or Jerad ever talk to me again. How about that? 😀

I think the one thing this thread very clearly does demonstrate is that there is a lot of stuff taught in schools, even at the best universities in the world, that is without a doubt just mental nonsense.

When you start taking mental concepts such as a “beginning”, or an “end”, or “unlimited”, or “all” or “large” , and then you pretend that you can make this into a definitive math problem, you are already just playing a game. There is no number for large, or a number for middle, or a number for vast or a number for time…these are things we imagine in our brains.

So when people start talking about how Cantor “proved” there are some infinities bigger than others, and this is taught as a reality in major universities, people have a right to be offended. It is an insult to the whole concept of knowledge. It would be like teaching at a university that red is beautiful, we have proved it. Or like teaching that time is frisbee shaped in heaven, we have proved it. And then middle school kids go around telling their friends that their brother taught them time was frisbee shaped. It can be proven mathematically.

No one can prove nonsense wrong, anymore than one can prove it is right. If someone was reading this thread hoping to see some insight as to why the mathematicians believe it is reasonable to side with Cantor, they will be sorely disappointed. The mathematicians are not using logic, or reason, or facts, they are just saying it is so, and hoping no one calls them on it. NO realities in life are solved by this math problem. NO new information about the world is gained by this absurd use of definitions. It makes a complete mockery of the concept of knowledge.

Once we see that you can make such absurd statements, such as those by Cantor, and then have it turned into a commonly taught believe, then everything else we are taught by supposedly trained professionals deserves to be second guessed-and at times ridiculed even. I think it undermines the whole profession of teaching. Cantors theory doesn’t teach us how to build a bridge stronger, or a better airplane, or how to find alternate energy sources. It teaches nothing. No platform of knowledge is made stronger by this imaginary card game.

This thread shows that a lot of things are just mental jokes like an emperor with no clothes that no one wants to mention.

Zachriel:

Funny since I just finished showing the opposite. There is no mathematics of infinity, otherwise my computer could not work with it. How many brains are contained in this “we” that you keep referring to? Are you a fruitcake or something? Or several fruitcakes?

phoodoo,

Do irrational numbers give you the vapors? What about complex numbers?

Keiths,

You still didn’t answer, do you believe in infinitely small points.

phoodoo:

The irony is that despite your rhetoric, it is the mathematicians who are being rigorous, while you, Mapou and Joe are succumbing to emotion and irrationality.

Cantor could teach you something about disciplined thinking if you would only listen.

Infinitely small points Keiths?

phoodoo #729,

Points are

definedas locations without extent. They are infinitely small, and there are infinitely many points between any two distinct points.How big do

youthink points are? Are they spheres, or do they have some other shape? How many per inch?keithS said.

Cantor could teach you something about disciplined thinking if you would only listen.

I say

amen.

I could not agree more,

from here

http://www.asa3.org/ASA/PSCF/1.....edman.html

quote:

During his university studies Cantor felt a deep calling from God to study philosophy and mathematics, rather than more lucrative pursuits. His faith sustained him during long years of rejection when the mathematical establishment dismissed his concept of the transfinite. When weaker men would have abandoned their work, Cantor perseve