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Why the universe cannot logically be infinite in time backwards

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File:Wooden hourglass 3.jpg
passage of time, imaged/S. Sepp

Further to “No Big Bang: Universe Always Was” just posted by Donald McLaughlin, Ashby Camp gave Uncommon Descent permission to post these notes from a class he taught at the 2018 Harding University Bible Lectureship titled “Answering the New Atheism.” Worth pondering:


1.The second premise of the Kalam cosmological argument is: The universe began to exist. It is more reasonable to believe this is true than to deny it because, Scripture aside, there are strong philosophical and scientific reasons for believing it.

a. The philosophical argument for the universe having a beginning is that past time cannot be infinite because an infinite amount of time cannot already have been exhausted so as to arrive at the present. Infinite time is limitless, inexhaustible, and thus cannot have been exhausted.

(1) Put differently, one could never traverse an infinite sequence of time units, an infinite number of seconds, minutes, hours, etc., to arrive at now. There always would be more time units to traverse before now. If one begins counting down from minus infinity, one cannot count to the present. An infinite amount of time can never pass because it is limitless; it can only be in process, never complete. To quote the New Dictionary of Christian Apologetics, (p. 700), “One can neither count from one to infinity nor count down from infinity to one. There is always an infinite distance to travel, so one never arrives.”

(2) Sean McDowell and Jonathan Morrow make the point this way in Is God Just a Human Invention? (Grand Rapids: Kregel Publications, 2010), 75-76:

Imagine you went for a walk in the park and stumbled across someone proclaiming aloud, “… five, four, three, two, one—there, I finally finished! I just counted down from infinity!” What would be your initial thought? Would you wonder how long the person had been counting? Probably not. More likely, you would be in utter disbelief. Why? Because you know that such a task cannot be done. Just as it’s impossible to count up to infinity from the present moment, it’s equally impossible to count down from … infinity to the present moment. Counting to infinity is impossible because there is always (at least) one more number to count. In fact, every time you count a number, you still have infinite more to go, and thus get no closer to your goal. Similarly, counting down from infinity to the present moment is equally impossible. Such a task can’t even get started! Any point you pick in the past to begin, no matter how remote, would always require (at least) one more number to count before you could start there. Any beginning point would require an infinite number of previous points. Here’s the bottom line: we could never get to the present moment if we had to cross an actual infinite number of moments in the past. Yet, since the present moment is real, it must have been preceded by a finite past that includes a beginning or first event. Therefore, the universe had a beginning.

(3) The impossibility of infinite past time, of having already traversed an infinite timespan, does not mean that future time will not go on forever. Future time is potentially not actually infinite. In other words, it is infinity in progress, something that will move toward infinity but never arrive; you’ll never get to the end of it. So it poses no problem like the claim of having already traversed an infinite timespan.

(4) The impossibility of traversing an infinite timespan, an infinite sequence of time units, need not mean that God has not always existed. There are ways of dealing with God’s relationship to time so that he does not exist (or has not always existed) in a sequence of individuated moments, a sequence of time units. His eternality is not one of infinite time but one of either timelessness or a different kind of time that has no measure or metric. For example, William Lane Craig’s view is that “God is timeless without creation and temporal since creation.” Philosophers Alan Padgett and Richard Swinburne refer to time before creation as “metrically amorphous time,” meaning it differs from our “measured time” (see, e.g., Eternity in Christian Thought).

(5) This philosophical claim that the universe cannot always have existed makes sense to me and to many philosophers, but there are others who are not persuaded.

(a) Some, for example, point to the fact a finite timespan, say one minute, can be subdivided infinitely into units of decreasing length, and yet one can still traverse that timespan. The claim is that in going from 0 to 1 minute one traverses an infinite number of time units to arrive at 1 minute, so it is not true that one cannot traverse an infinite number of time units to arrive at the present. But there is a problem with that claim. The subdivisions of a finite timespan are only potentially infinite in number. It is true that one could keep subdividing forever, but each further subdivision results in a finite number of subdivisions the sum of which is the finite timespan being subdivided. The number of subdivisions can grow toward an infinite number but can never actually reach an infinite number. Whereas, when speaking of an infinite timespan one is speaking of an actually infinite set of time units, the sum of which is an infinite length of time.

(b) Though this philosophical claim that the universe necessarily began to exist continues to be debated, it dovetails nicely with the scientific acceptance of the universe having a beginning. That is the subject to which I now turn.

  

Readers?

Comments
ET, 57: That doesn’t follow. I would expect all countably infinite sets to be able to show bijection. You can show a bijection between any pair of countably infinite sets. If you can't then one is not countably infinite. Also the positive integers already have a derived/ natural match with the even integers. Yours need to contrive a matching formula. Doesn't matter, if you can find one they are the same size. That's the key. Any one-for-one matches shows the sets are the same size. Sometimes there's more than one one-for-one matching but any one will do.JVL
March 9, 2020
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ET, 56: That is one of the most stupid things that I have ever read. At least with my method set theory, that foundation of mathematics, is consistent from bottom to top. Then how is it I can match the set of positive integers one-for-one with the set of positive even integers? If one set is bigger I shouldn't be able to do that but I can. No one uses the concept under discussion. No one. That means no one, Jerad. Not you nor anyone else. You're not a mathematicians so you are unaware of its use. Infinite sets work exactly as I have said. And Jerad cannot demonstrate otherwise. So what does the coward do? Throw out moar asinine innuendos. People see what you are doing, Jerad. No one is fooled. If anyone wants to look it up they will find out who is correct. No mathematician uses relative cardinalities. Not one. Because there's no point: all countably infinite sets are the same size. And if you disagree with Cantor's work then please, by all means, point out a flaw. With my methodology we have consistency from the bottom to the top. We have the SAME matching criteria. We have the SAME set subtraction methodology. And, where applicable, the bijection formula becomes the relative cardinality. If you want to propose a different system then do some work and publish it. See if anyone likes it. And guess what? Bridges won’t fail. The Moon’s orbital dynamics won’t change. And it will give mathematicians something to do- find the relative cardinalities of all countably infinite sets. The mathematicians are plenty busy already. And it's still true that there are just as many positive even integers as there are positive integers. But Jerad prefers to live in the 19th century. In mathematics true things stay true. The Pythagorean theorem was proven over 2000 years ago and is still true. Newton and Leibnitz invented Calculus and it's still true. I'm very happy to stick with stuff that's true no matter how old it is.JVL
March 9, 2020
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JVL:
If there are more positive integers than positive even integers then it should be impossible to match them up one-for-one so that each element of each set has a unique partner in the other set.
That doesn't follow. I would expect all countably infinite sets to be able to show bijection. Also the positive integers already have a derived/ natural match with the even integers. Yours need to contrive a matching formula.ET
March 9, 2020
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JVL:
Your “set subtraction” doesn’t verify it because “set subtraction” doesn’t apply for infinite sets.
That is one of the most stupid things that I have ever read. At least with my method set theory, that foundation of mathematics, is consistent from bottom to top. No one uses the concept under discussion. No one. That means no one, Jerad. Not you nor anyone else. Infinite sets work exactly as I have said. And Jerad cannot demonstrate otherwise. So what does the coward do? Throw out moar asinine innuendos. People see what you are doing, Jerad. No one is fooled. With my methodology we have consistency from the bottom to the top. We have the SAME matching criteria. We have the SAME set subtraction methodology. And, where applicable, the bijection formula becomes the relative cardinality. And guess what? Bridges won't fail. The Moon's orbital dynamics won't change. And it will give mathematicians something to do- find the relative cardinalities of all countably infinite sets. But Jerad prefers to live in the 19th century.ET
March 9, 2020
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If there are more positive integers than positive even integers then it should be impossible to match them up one-for-one so that each element of each set has a unique partner in the other set. But you can find such a pairing so the sets must be the same size. It's the same with any pair of infinite sets: if you can match the elements up one-for-one then the sets must be the same size. That's what Cantor realised and it works for finite as well as infinite sets.JVL
March 9, 2020
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ET, 51: They are your opinions. Set subtraction proves that what you are saying is nonsense. So you ignore it. Your "set subtraction" does not apply. You want infinite sets to work the same as finite sets and they don't. No one uses the concept under discussion. No one. Just because you don't doesn't mean no one does. Cantor didn’t prove anything. My arguments are spot on. My example proves that Jerad cannot think. Can you find a fault with Cantor's proofs? And as predicted Jerad retreats to “set theory”. You are a coward, Jerad. And a predictable coward at that. Set Theory is an important and foundation area of mathematics. You can take classes just on Set Theory. If all countably infinite sets had the same cardinality then set subtraction should verify that. Yet it shows the opposite. Your "set subtraction" doesn't verify it because "set subtraction" doesn't apply for infinite sets. Infinite sets work exactly as I have said. And Jerad cannot demonstrate otherwise. I don't want to be rude but no mathematician agrees with you because they understand the work that Cantor did was correct. Bijection just proves that both sets are countable. Set subtraction can prove that the number of elements is not the same. Still incorrect. All countably infinite sets have the same cardinality.JVL
March 9, 2020
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What Cantor realised was that IF two sets have the same number of elements then you can match up all the elements from both sets one-for-one, i.e. each element of the first set is matched with one and only one element of the second set and the same in the other direction. So, IF you can show a way to match up two sets in that way then they are the same size. This works for finite and infinite sets. If you can't perform such a matching then the sets must be of different sizes or cardinalities. So, for each positive integer j in the set of all positive integers match it up with 2 x j in the set of positive even integers, the second set. Each positive integer is matched with one and only one even integer in the second set and each element in the second set is matched with one and only one element in the first set. No element of either set is left out. This can only happen if both sets have the same number of elements. So the sets are the same size.JVL
March 9, 2020
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Bijection just proves that both sets are countable. Set subtraction can prove that the number of elements is not the same.ET
March 9, 2020
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They are your opinions. Set subtraction proves that what you are saying is nonsense. So you ignore it. No one uses the concept under discussion. No one. Cantor didn't prove anything. My arguments are spot on. My example proves that Jerad cannot think. And as predicted Jerad retreats to "set theory". You are a coward, Jerad. And a predictable coward at that. If all countably infinite sets had the same cardinality then set subtraction should verify that. Yet it shows the opposite. Infinite sets work exactly as I have said. And Jerad cannot demonstrate otherwise.ET
March 9, 2020
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Just in case someone wants an actual proof that the cardinality of the positive even integers is the same as the positive integers you can find it here along with some other proofs. https://courses.cs.cornell.edu/cs2800/wiki/index.php/FA18:Lecture_8_cardinalityJVL
March 9, 2020
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Correction in 47 above, last line: "unpins" should be "underpins" but "supports" would have been a better descriptive actually.JVL
March 9, 2020
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ET, 45: Jeard, your opinions are full of crap. And all you have are your opinions. They're not my opinions. What I am saying is correct. No one uses it. Mathematicians use it. And, by transference, many, many other sciences. Your opinion, which stinks. Not an opinion. What I am saying is correct. Set subtraction can prove that one countably infinite set has more elements than another. And all Jerad can do is reference a dead guy who didn’t understand relativity. All countably infinite sets have the same cardinality. Cantor proved it. I can explain the reasoning if anyone wants me to. Or you can look it up, the truth of my statements remains. It is very telling that Jerad refuses to answer my questions and respond to my arguments. Your arguments are incorrect, infinite sets do not work they way you think they do. I have established that Cantor's set theory is considered fundamental to modern mathematics by many mathematicians for different parts of mathematics at least. It's like the foundation of a building which surely is very useful. Everything I say can be verified by doing a bit of work and searching online.JVL
March 9, 2020
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From: https://en.wikipedia.org/wiki/Set_theory
Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.[note 1] The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose ontology consists of: Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC). Fragments of ZFC include: Zermelo set theory, which replaces the axiom schema of replacement with that of separation; General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets; Kripke–Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the axiom schemata of separation and replacement. Sets and proper classes. These include Von Neumann–Bernays–Gödel set theory, which has the same strength as ZFC for theorems about sets alone, and Morse–Kelley set theory and Tarski–Grothendieck set theory, both of which are stronger than ZFC. The above systems can be modified to allow urelements, objects that can be members of sets but that are not themselves sets and do not have any members. The New Foundations systems of NFU (allowing urelements) and NF (lacking them) are not based on a cumulative hierarchy. NF and NFU include a "set of everything, " relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of classical logic. Yet other systems accept classical logic but feature a nonstandard membership relation. These include rough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relation is not simply True or False. The Boolean-valued models of ZFC are a related subject. An enrichment of ZFC called internal set theory was proposed by Edward Nelson in 1977.
That's the real stuff.
Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice.[1] Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes human-written, computer-verified derivations of more than 12,000 theorems starting from ZFC set theory, first-order logic and propositional logic.
There's lots of stuff on countability here: https://en.wikipedia.org/wiki/Countable_set Remember that the integers are a countably infinite set. The most significant contributions to the field were made by Georg Cantor, a genius of the first order. From his article:
Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite.[1] This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article was published in 1874. In 1879, he modified his uncountability proof by using the topological notion of a set being dense in an interval. Cantor's article also contains a proof of the existence of transcendental numbers. As early as 1930, mathematicians have disagreed on whether this proof is constructive or non-constructive. Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved. Since Cantor's proof either constructs transcendental numbers or does not, an analysis of his article can determine whether his proof is constructive or non-constructive. Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that he had a choice between two proofs, one that uses the uncountability of the real numbers and one that does not. Historians of mathematics have examined Cantor's article and the circumstances in which it was written. For example, they have discovered that Cantor was advised to leave out his uncountability theorem in the article he submitted?—?he added it during proofreading. They have traced this and other facts about the article to the influence of Karl Weierstrass and Leopold Kronecker. Historians have also studied Dedekind's contributions to the article, including his contributions to the theorem on the countability of the real algebraic numbers. In addition, they have recognized the role played by the uncountability theorem and the concept of countability in the development of set theory, measure theory, and the Lebesgue integral.
The distinction between countable and uncountable sets is key and a very, very obvious side note in the whole discussion is the fact that the positive integers and the positive even integers have the same cardinality. That fact is all part of a class of work that many mathematicians feel is fundamental to modern mathematics or at least some parts. That's why it's useful; it unpins a lot of other work.JVL
March 9, 2020
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It is very telling that Jerad refuses to answer my questions and respond to my arguments.ET
March 9, 2020
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Jeard, your opinions are full of crap. And all you have are your opinions.
Just because you don’t use it or understand the uses doesn’t mean it’s useless or false.
No one uses it.
Look up Set Theory as in Cantor’s set theory.
I have.
Still incorrect.
Your opinion, which stinks.
It’s useful to mathematicians.
Liar. Set subtraction can prove that one countably infinite set has more elements than another. And all Jerad can do is reference a dead guy who didn't understand relativity.ET
March 9, 2020
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ET, 42: Ad yet all you do is try to insult me. It's just a statement of fact. Your statements clearly indicate you don't understand how set theory works beyond the basic simple operations. I repeat because it is right and there isn’t anything you can do or say. I don't have to do or say anything: what you say is incorrect as can be easily verified. And if a concept isn’t useful that means it is meaningless. And that means anyone can say anything and nothing changes. Just because you don't use it or understand the uses doesn't mean it's useless or false. The concept that all countably infinite sets have the same cardinality is useless. Meaning it doesn’t have any practical value. No one uses it for anything. And Jerad has NEVER found anything that says otherwise. Look up Set Theory as in Cantor's set theory. Jerad doesn’t understand infinity, and it shows. Jerad doesn’t understand sets, and it shows. A simple online search will show who is correct. And we are right back to the fact that saying two countably infinite sets having the same cardinality is of no use. That means it is meaningless to say such a thing. It also means one has to lie when set subtraction proves that one countably infinite set has more elements than another. Still incorrect. Please, by all means, look it up. I challenge any and every one to find a practical use for saying that all countably infinite sets have the same cardinality. It's useful to mathematicians. And it's true as well.JVL
March 9, 2020
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And we are right back to the fact that saying two countably infinite sets having the same cardinality is of no use. That means it is meaningless to say such a thing. It also means one has to lie when set subtraction proves that one countably infinite set has more elements than another. Please, by all means, look it up. I challenge any and every one to find a practical use for saying that all countably infinite sets have the same cardinality.ET
March 9, 2020
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Jerad:
Being insulting doesn’t make you right
Ad yet all you do is try to insult me.
Repeating over and over again doesn’t make it right.
I repeat because it is right and there isn't anything you can do or say. And if a concept isn't useful that means it is meaningless. And that means anyone can say anything and nothing changes. The concept that all countably infinite sets have the same cardinality is useless. Meaning it doesn’t have any practical value. No one uses it for anything. And Jerad has NEVER found anything that says otherwise. Jerad doesn't understand infinity, and it shows. Jerad doesn't understand sets, and it shows.ET
March 9, 2020
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ET, 39: Of course it is. That is what the … means- more of the same. Are you really that daft? Look it up, Jerad. Or shut up. I know what infinity means in mathematics. I've worked with in in Calculus, Analysis, Set Theory, Complex Analysis, Statistics, etc. IF the set of all positive even integers has the same cardinality as the set of all positive integers, THEN set subtraction would not be able to prove otherwise. And yet it does. Repeating over and over again doesn't make it right. Jerad can whine all he wants but what I said is a fact. Nope. Look it up. OR: find some reference that agrees with you. Take the set of all positive integers and subtract the set of all evens. If the two sets had the same cardinality the answer would be 0. And yet the answer is the set of all odd integers. Yup, which has the same cardinality of the evens and the integers. And the rational numbers by the way. Cardinality refers to the number of elements in a set. Therefore if set subtraction comes back with something other than nothing the cardinalities cannot be the same. math 101. Not with infinite sets. Look it up. And we are right back to the fact that saying two countably infinite sets having the same cardinality is of no use. That means it is meaningless to say such a thing. It also means one has to lie when set subtraction proves that one countably infinite set has more elements than another. You can repeat yourself all you want, what you are saying is still not correct. Look it up. I looked it up and no one uses that concept for anything. All Jerad can do is lie and whine. I'll leave it up to anyone else reading this to look it up for themselves and see. I understand how set theory works, Jerad. Clearly you cannot think for yourself. Being right is okay with me. “Things are different for infinity” comes from the little minds that cannot grasp the concept. Still stands. Being insulting doesn't make you right. And we are right back to the fact that saying two countably infinite sets having the same cardinality is of no use. That means it is meaningless to say such a thing. It also means one has to lie when set subtraction proves that one countably infinite set has more elements than another. Repeating the same thing over and over again doesn't make it correct. The concept that all countably infinite sets have the same cardinality is useless. Meaning it doesn’t have any practical value. No one uses it for anything. And Jerad has NEVER found anything that says otherwise. Useful or not it's still correct, which you can easily verify by looking it up.JVL
March 9, 2020
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And we are right back to the fact that saying two countably infinite sets having the same cardinality is of no use. That means it is meaningless to say such a thing. It also means one has to lie when set subtraction proves that one countably infinite set has more elements than another. The concept that all countably infinite sets have the same cardinality is useless. Meaning it doesn't have any practical value. No one uses it for anything. And Jerad has NEVER found anything that says otherwise.ET
March 9, 2020
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JVL:
Nope, infinity is not just more of the same.
Of course it is. That is what the … means- more of the same. Are you really that daft? Look it up, Jerad. Or shut up. IF the set of all positive even integers has the same cardinality as the set of all positive integers, THEN set subtraction would not be able to prove otherwise. And yet it does. Jerad can whine all he wants but what I said is a fact. Take the set of all positive integers and subtract the set of all evens. If the two sets had the same cardinality the answer would be 0. And yet the answer is the set of all odd integers. Cardinality refers to the number of elements in a set. Therefore if set subtraction comes back with something other than nothing the cardinalities cannot be the same. math 101. And we are right back to the fact that saying two countably infinite sets having the same cardinality is of no use. That means it is meaningless to say such a thing. It also means one has to lie when set subtraction proves that one countably infinite set has more elements than another. I looked it up and no one uses that concept for anything. All Jerad can do is lie and whine.
I’m not the one that doesn’t understand how set theory works.
I understand how set theory works, Jerad. Clearly you cannot think for yourself. “Things are different for infinity” comes from the little minds that cannot grasp the concept. Still stands.ET
March 9, 2020
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ET: 37: Jerad, you are a liar. That you have to use false accusations, seals the deal. Reality escapes you. The ability to think for yourself, escapes you. Only a moron thinks that something special happens with infinity. And here you are. Infinity is just more of the same. Nope, infinity is not just more of the same. You don't have to believe me, look it up! Are you afraid to find out I'm right? IF the set of all positive even integers has the same cardinality as the set of all positive integers, THEN set subtraction would not be able to prove otherwise. And yet it does. Nope, that's not right. Look it up if you don't believe me. Take the set of all positive integers and subtract the set of all evens. If the two sets had the same cardinality the answer would be 0. And yet the answer is the set of all odd integers. Yup, and all three sets have the same cardinality. Look it up. It's easy to look up. And we are right back to the fact that saying two countably infinite sets having the same cardinality is of no use. That means it is meaningless to say such a thing. It also means one has to lie when set subtraction proves that one countably infinite set has more elements than another. Look it up. You'll never believe me so look it up. And all Jerad can do is whine like a little baby. Being right is not whining. “Things are different for infinity” comes from the little minds that cannot grasp the concept. I'm not the one that doesn't understand how set theory works. All you have to do is look up the pertinent mathematics to see I'm correct. It's easy.JVL
March 9, 2020
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Jerad, you are a liar. That you have to use false accusations, seals the deal. Reality escapes you. The ability to think for yourself, escapes you. Only a moron thinks that something special happens with infinity. And here you are. Infinity is just more of the same. IF the set of all positive even integers has the same cardinality as the set of all positive integers, THEN set subtraction would not be able to prove otherwise. And yet it does. Take the set of all positive integers and subtract the set of all evens. If the two sets had the same cardinality the answer would be 0. And yet the answer is the set of all odd integers. And we are right back to the fact that saying two countably infinite sets having the same cardinality is of no use. That means it is meaningless to say such a thing. It also means one has to lie when set subtraction proves that one countably infinite set has more elements than another. And all Jerad can do is whine like a little baby. "Things are different for infinity" comes from the little minds that cannot grasp the concept.ET
March 9, 2020
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<b<ET: Bring back the edit feature… Please Something we can definitely agree on!! Who thought taking it away was a good idea?JVL
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ET, 32: Here we go. Pound sand, Jerad. Thank you for proving that you are a coward. All you can do is hurl dales accusations. That makes you a punk, too. It's true, you don't understand set theory. I mean real set theory beyond Algebra 101. Sets are COLLECTIONS of things. My examples are of things being collected. My example has one counter always and forever having more than the other. At EVERY finite point in time, into infinity. Things are different for infinity. Look it up. You can’t even read for comprehension. Whatever. You don't understand set theory. IF the set of all positive even integers has the same cardinality as the set of all positive integers, THEN set subtraction would not be able to prove otherwise. And yet it does. Nope, it doesn't. The set of all positive integers, the set of all positive even integers, the set of all positive odd integers, the set of all positive multiples of three, the set of all positive prime numbers, the set of all positive powers of 4 all have the same cardinality. The set of all real numbers has a different cardinality. Look it up. That is a FACT. Nope. Look it up. It doesn't matter what you think, what matters is what is true. Look it up. And we are right back to the fact that saying two countably infinite sets having the same cardinality is of no use. That means it is meaningless to say such a thing. It also means one has to lie when set subtraction proves that one countably infinite set has more elements than another. I'm sorry the math escapes you. You're not the only one if that makes you feel better.JVL
March 9, 2020
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hurl FALSE accusations- Bring back the edit feature... PleaseET
March 9, 2020
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And we are right back to the fact that saying two countably infinite sets having the same cardinality is of no use. That means it is meaningless to say such a thing. It also means one has to lie when set subtraction proves that one countably infinite set has more elements than another.ET
March 9, 2020
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Here we go. Pound sand, Jerad. Thank you for proving that you are a coward. All you can do is hurl dales accusations. That makes you a punk, too. Sets are COLLECTIONS of things. My examples are of things being collected. My example has one counter always and forever having more than the other. At EVERY finite point in time, into infinity. You can't even read for comprehension. IF the set of all positive even integers has the same cardinality as the set of all positive integers, THEN set subtraction would not be able to prove otherwise. And yet it does. That is a FACT.ET
March 9, 2020
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ET, 29: My questions have everything to do with the cardinality of infinite sets. And your responses prove that you do not understand that infinity is a journey. You don't understand set theory. Cardinality has to work for ALL sets. My example pertains to two specific sets. The accepted version does work for all sets. You can't even figure out the cardinality of some sets. Like the primes; what's the cardinality of the primes? IF the set of all positive even integers has the same cardinality as the set of all positive integers, THEN set subtraction would not be able to prove otherwise. And yet it does. Nope, it doesn't. If you take the set of the positive integers and "subtract" the set of positive even integers you get the set of positive odd integers and all three of those sets have the same cardinality. It's easy. Jerad won’t answer my questions because if he did then he would show that my premise is correct. And I am more than OK with that. You don't understand set theory. If you count in the manner you describe and you stop at some time then one counter would be twice as big as the other. But that wouldn't be at infinity would it? You're questions have nothing to do with how set theory works. Look it up. It's easy to find out the way it works.JVL
March 9, 2020
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Jerad won't answer my questions because if he did then he would show that my premise is correct. And I am more than OK with that. Thanks, Jerad.ET
March 9, 2020
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