Are some infinities bigger than others?

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February 15, 2018

Cosmology, Intelligent Design, Mathematics, Philosophy

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Intelligent Design
Mathematics
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From theoretical physicist Paul Davies at Cosmos:

It turns out that the set of all points on a continuous line is a bigger infinity than the natural numbers; mathematicians say there is an uncountably infinite number of points on the line (and in three-dimensional space). You simply can’t match up each point on the line with the natural numbers in a one-to-one correspondence.

…

If it is continuous (and some physicists think it may not be) then it will contain an uncountably infinite number of points. But that doesn’t mean it has to go on forever. As Einstein discovered, it may be curved in on itself to form a finite volume. More.

One would have thought that once we get involved with infinities, we are skirting the boundaries of measurement—at best.

See also: God as a necessary, maximally great, endless being vs. the challenge to an actual infinity

Reader: Weirdness of infinity shows that the universe is not infinitely old

Durston and Craig on an infinite temporal past . . .

At Quanta: Is infinity real?

and

Cosmologist: In an infinite multiverse, physics loses its ability to make predictions. And that’s okay.

Comments

H'mm, I just thought of a way to drive the matter home conceptually. Not a proof, an illustration. Let any given reciprocal be 1/n, from 1, 2, 3 . . . then do the set for the interval (1,2] by adding 1 +1/n in succession, i.e. 1 +1, 1 + 1/2, 1 + 1/3 etc. The same dust reappears in (1,2]. Then, do it for 2 + 1/n, 3 + 1/n etc. Copies of the dust appear in the intervals between all counting numbers. And in no case does it begin to exhaust the interval (n, n+1]. If that does not astonish you, it should. KF PS: try the von Neumann construction of the naturals: {} --> 0 {0} --> 1 {0,1} --> 2 etc, endlessly. Now recognise that we have a new class of quantity, the first transfinite, of cardinality aleph null. Then go on from there. There are transfinitely many transfinites as the surreals tell us.kairosfocus_{February 15, 2018
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News, Davies is demonstrably right, C exceeds aleph-null, as can be shown. For that matter, the reciprocal of every counting number from 1 up lies in the interval (0,1], as a dust, these are not connected points. The vast bulk, in the close neighbourhood of zero -- not a proof but an illustration. So a dual to the set of naturals is found in the continuum between 0 and 1. Astonishing, if you ponder it. AS: The numbers apply to different aspects. In orthodox Christian theology, God is unified as to being, diverse as to person; thus, triune. Infinity as applied to the Godhead has to do with being the greatest, beyond finite bound. But at the same time God cannot make a square circle, that is impossible of being. Likewise God will not violate his core character so will be truthful, etc. KFkairosfocus_{February 15, 2018
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you may be persuaded that this has any meaning spiritual or physical
JDH, Infinity as Theology is an entirely different matter. But then you have to throw the numbers (as we can use them) away, if you go there. I don't think mathematics apply. I think mathematics are a tool given to us finite creatures by God. You get mysteries like 3 and 1 at the same time when you think about God in a numerical context. No calculator has that function. Andrewasauber_{February 15, 2018
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asauber@12 That's an interesting question. As many have noted in the field of mathematics intuition from FINITE sets CANNOT answer questions about INFINITE sets. Thus when one studies infinities one comes up with things which are finite minds have no intuition for. Thus it is easy to prove that two countable infinite sets have the EXACT same number of elements. Mathematicians do this by proving there is a 1-1 mapping from every element of one set to an element in the other set. So for example consider the natural numbers (counting numbers) starting from 1. ( 1, 2, 3, 4, 5..) And the set of counting numbers starting from 2. ( 2, 3, 4, 5...) Using the intuition from finite sets one could reason that the first set has every element that the second set does AND the number 1 (one). Since the first set has every number the second set does + 1 more, finite reasoning (i.e. reasoning about finite sets) would conclude the first set has more elements. This finite reasoning would be dead wrong. Infinity as a concept does not become bigger by adding a finite number to it. There is a simple mapping ( n2 = n1 + 1) for each n1 in the first set that makes a 1-1 correspondence between the two sets. This is what is meant that they have equal cardinality or in layman's terms are the same size infinity. Cantor showed that there is no way to build a map between the counting numbers and the set of real numbers ( points on the number line ). So it is true that in mathematical reasoning the set of real numbers is somehow bigger. The interesting question you raise is as to whether this is any more than a mathematical exercise. I will leave the question of whether you may be persuaded that this has any meaning spiritual or physical to the ample list of links shared by BA77JDH_{February 15, 2018
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Infinity is infinity. The alleged difference is how many elements each numerical infinity contain.ET_{February 15, 2018
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DaveS, To be straightforward in letting you know where I'm coming from, someone is going to have to produce evidence that Infinity is more than just a entertaining mental exercise. I have only so much time and mental space to use, so I try to make all my exercising worth the while, whether it be physical excercising, mental or spiritual. Andrewasauber_{February 15, 2018
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You might discover why people say that some infinities are bigger than others.
You're digging the hole deeper. Andrewasauber_{February 15, 2018
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Albeit in a somewhat different context, William James, riffing on Bergson, once said this about the nature of mathematical mind: 'When the mathematician,’ Bergson writes, ‘calculates the state of a system at the end of a time t, nothing need prevent him from supposing that betweenwhiles the universe vanishes, in order suddenly to appear again at the due moment in the new configuration. It is only the t-th moment that counts—that which flows throughout the intervals, namely real time, plays no part in his calculation…. In short, the world on which the mathematician operates is a world which dies and is born anew at every instant, like the world which Descartes thought of when he spoke of a continued creation.’ To know adequately what really happens we ought, Bergson insists, to see into the intervals, but the mathematician sees only their extremities. He fixes only a few results, he dots a curve and then interpolates, he substitutes a tracing for a reality. http://www.philosophy-index.com/james/pluralistic-universe/vi.php Multiple infinities? Could all those countless "betweenwhiles" perhaps just be so many figments of the limited, dividing mind? I'd prefer to trust the perennial sages who say "One being, without a second".BPS from AZ_{February 15, 2018
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Of personal note: I hold it to be fairly obvious that 'growing large without measure' is a lesser quality infinity than a fraction in which the denominator goes to zero.
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
Simply put, the reason why 'growing large without measure' is a lesser quality infinity than 'a fraction in which the denominator goes to zero' is because anything that begins to grow large without measure must necessarily have some sort of beginning in time and must also necessarily have some sort of preexistent space to grow into. Whereas, on the other hand, to form a fraction in which the denominator goes to zero is to force a finite object into a type of infinity that can have no discernible beginning in space or time. That is to say, since zero represents the absence of everything including the absence of space and time, then to form a fraction in which the denominator goes to zero is to force a finite object into a true infinity that is beyond space and time. In more technical terms, I view growing large without measure to be a potential infinite and I view a fraction in which the denominator goes to zero to be an actual infinite.
Potential Infinity vs. Actual Infinity - June 7, 2012 by Ryan Excerpt: In a potential infinity, one can keep adding or subdividing without end, but one never actually reaches infinity. In a sense, a potential infinity is an endless process that at any point along the way is finite. By contrast, in an actual infinity, the infinite is viewed as a completed totality. http://www.numbersleuth.org/trends/potential-vs-actual-infinity/
Moreover, if we rightly let the Agent causality of God ‘back’ into the picture of modern physics, as the Christian founders of modern science originally envisioned, (Newton, Maxwell, Faraday, and Planck, to name a few), , then an empirically backed reconciliation between Quantum Mechanics and General Relativity, i.e. the ‘Theory of Everything’, readily pops out for us in Christ’s resurrection from the dead. Specifically, the fact that Jesus Christ dealt with both general relativity and quantum mechanics in His resurrection from the dead is made evident by the Shroud of Turin.
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. http://www.academicjournals.org/sre/PDF/pdf2012/30JulSpeIss/Antonacci.pdf The absorbed energy in the Shroud body image formation appears as contributed by discrete (quantum) 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/mus/541/1/c1a0802004.pdf Astonishing discovery at Christ's tomb supports Turin Shroud - NOV 26TH 2016 Excerpt: The first attempts made to reproduce the face on the Shroud by radiation, used a CO2 laser which produced an image on a linen fabric that is similar at a macroscopic level. However, microscopic analysis showed a coloring that is too deep and many charred linen threads, features that are incompatible with the Shroud image. Instead, the results of ENEA “show that a short and intense burst of VUV directional radiation can color a linen cloth so as to reproduce many of the peculiar characteristics of the body image on the Shroud of Turin, including shades of color, the surface color of the fibrils of the outer linen fabric, and the absence of fluorescence”. 'However, Enea scientists warn, "it should be noted that the total power of VUV radiations required to instantly color the surface of linen that corresponds to a human of average height, body surface area equal to = 2000 MW/cm2 17000 cm2 = 34 thousand billion watts makes it impractical today to reproduce the entire Shroud image using a single laser excimer, since this power cannot be produced by any VUV light source built to date (the most powerful available on the market come only to several billion watts)”. Comment The ENEA study of the Holy Shroud of Turin concluded that it would take 34 Thousand Billion (trillion) Watts of VUV radiation to make the image on the shroud. This output of electromagnetic energy remains beyond human technology. https://www.ewtn.co.uk/news/latest/astonishing-discovery-at-christ-s-tomb-supports-turin-shroud
Verses
Matthew 28:18 And Jesus came to them and spake unto them, saying, All authority hath been given unto me in heaven and on earth. Colossians 1:15-20 The Son is the image of the invisible God, the firstborn over all creation. For in him all things were created: things in heaven and on earth, visible and invisible, whether thrones or powers or rulers or authorities; all things have been created through him and for him. He is before all things, and in him all things hold together. And he is the head of the body, the church; he is the beginning and the firstborn from among the dead, so that in everything he might have the supremacy. 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.
This following video also touches on the 'higher dimensional' math behind Quantum Mechanics, Special Relativity and General Relativity
Quantum Mechanics, Special Relativity, General Relativity and Christianity https://www.youtube.com/watch?v=gKggH8jO0pk
bornagain77_{February 15, 2018
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In the following video, Richard Feynman rightly expresses his unease with “brushing infinity under the rug.” in Quantum-Electrodynamics:
“It always bothers me that in spite of all this local business, what goes on in a tiny, no matter how tiny, region of space, and no matter how tiny a region of time, according to laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out. Now how can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do?" - Richard Feynman – one of the founding fathers of QED (Quantum Electrodynamics) Quote taken from the 6:45 minute mark of the following video: Feynman: Mathematicians versus Physicists - video https://www.youtube.com/watch?v=obCjODeoLVw
I don’t know about Richard Feynman, but as for myself, as a Christian Theist, I find it rather comforting to know that it takes an ‘infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do’:
“Why should it take an infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do?" - Richard Feynman John1:1 "In the beginning was the Word, and the Word was with God, and the Word was God." of note: ‘the Word’ in John1:1 is translated from ‘Logos’ in Greek. Logos is also the root word from which we derive our modern word logic http://etymonline.com/?term=logic
,,, And whereas special relativity, by ‘brushing infinity under the rug’, has been successfully unified with quantum theory to produce Quantum Electrodynamics, no such mathematical ‘sleight of hand’ exists for unifying general relativity with quantum mechanics. General relativity simply refuses to be mathematically unified with quantum mechanics in any acceptable way. In technical terms, Gravity has yet to be successfully included into a theory of everything since the infinities that crop up in that attempt are not renormalizable as they were in Quantum-Electrodynamics.
Unified field theory Excerpt: Gravity has yet to be successfully included in a theory of everything. Simply trying to combine the graviton with the strong and electroweak interactions runs into fundamental difficulties since the resulting theory is not renormalizable. Theoretical physicists have not yet formulated a widely accepted, consistent theory that combines general relativity and quantum mechanics. The incompatibility of the two theories remains an outstanding problem in the field of physics. Some theoretical physicists currently believe that a quantum theory of general relativity may require frameworks other than field theory itself, such as string theory or loop quantum gravity. https://en.wikipedia.org/wiki/Unified_field_theory#Current_status Does quantum mechanics contradict the theory of relativity? Sanjay Sood, Microchip Design Engineer, Theoretical and Applied Physicist - Feb 14, 2016 Excerpt: quantum mechanics was first integrated with special theory of relativity by Dirac in 1928 just 3 years after quantum mechanics was discovered. Dirac produced an equation that describes the behavior of a quantum particle (electron). In this equation the space and time enter on the same footing - equation is first order in all 4 coordinates. One startling by product of this equation was the prediction of anti matter. It also gave the correct explanation for the electron's spin. Dirac's equation treats an electron as a particle with only a finite degrees of freedom. In 1940s Dirac's equation was incorporated into the relativistic quantum field theory that's knowns as quantum electrodynamics (QED) independently by Feynman, Schwinger and Tomonaga. This is the theory that describes the behavior of electrons and photons and their interactions with each other in terms of relativistic quantum fields that have infinite degrees of freedom. QED allowed extremely precise calculation of anomalous magnetic dipole moment of an electron. This calculated value matches the experimentally measured value to an astonishing precision of 12 decimal places! The integration of Einstein's general theory of relativity and quantum mechanics has proved to be far more difficult. Such an integration would give a quantum theory of gravity. Even after a sustained effort lasting more than half a century, no renormalized quantum field theory of gravity has ever been produced. Renormalization means a theory that's free of infinities at zero distance or infinite energy because 2 point particles can interact with each other at zero distance. A non renormalizable theory has no predictive value because it contains an infinite number of singular coefficients. https://www.quora.com/Does-quantum-mechanics-contradict-the-theory-of-relativity
Some theoretical physicists have remarked that this failure to mathematically unify Quantum Mechanics and General Relativity into a mathematical “Theory of Everything” is ‘the collapse of physics as we know it’ At the 4:30 minute mark of the following video, Michio Kaku, after going through some calculations with the equation of General Relativity showing how it breaks down at black holes, states that, 'It means the collapse of everything we know about the physical universe.'
“Here is the problem (with black holes), right there, when 'r' (radius) is equal to zero, The point at which physics itself breaks down. So 1 over 'r' equals 1 over 0 equals infinity. To a mathematician infinity is simply a number without limit. To a physicist it is a monstrosity. It means first of all that gravity is infinite at the center of a black hole. That time stops. And what does that mean? Space makes no sense. It means the collapse of everything we know about the physical universe. In the real world there is no such thing as infinity. Therefore there is a fundamental flaw in the formulation of Einstein's theory.” Quantum Mechanics & Relativity – Michio Kaku - The Collapse Of Physics As We Know It ? - video http://www.dailymotion.com/video/x2jbd7x
Then in the same video at the 7:08 minute mark, Michio Kaku,, after going through some calculations trying to integrate Quantum Mechanics and Gravity at a black hole, goes on to state “And when you do this integral, you get something which makes no sense whatsoever. An infinity. Total nonsense. In fact, you get an infinite sequence of infinities. Infinitely worse than the divergences of Einstein's original theory.”
“And when you do this integral, you get something which makes no sense whatsoever. An infinity. Total nonsense. In fact, you get an infinite sequence of infinities. Infinitely worse than the divergences of Einstein's original theory.” Quantum Mechanics & Relativity – Michio Kaku - The Collapse Of Physics As We Know It ? - video http://www.dailymotion.com/video/x2jbd7x
As the preceding video clearly illustrated, the main conflict of reconciling General Relativity and Quantum Mechanics appears to arise from the inability of either theory to successfully deal with the Zero/Infinity conflict that crops up between 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/edit01_02/edit6_mar02.htm
Yet, the unification, into a 'theory of everything', between what is in essence the 'infinite Theistic world of Quantum Mechanics' and the 'finite Materialistic world of the space-time of General Relativity' seems to be directly related to what Jesus apparently joined together with His resurrection, That is to say, it looks to be directly related to the unification of infinite God with finite man. Dr. William Dembski in this following comment, although he was not addressing the Zero/Infinity conflict in General Relativity and Quantum Mechanics, offers this insight into what the 'unification' of infinite God and the finite man might look like mathematically:
The End Of Christianity - Finding a Good God in an Evil World - Pg.31 William Dembski PhDs. Mathematics and Theology 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/documents/2009.05.end_of_xty.pdf Philippians 2:8-9 And being found in appearance as a man, He humbled Himself and became obedient to the point of death, even the death of the cross. Therefore God also has highly exalted Him and given Him the name which is above every name,
bornagain77_{February 15, 2018
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Gödel, Infinity, and Jesus Christ as the Theory of Everything - video https://www.youtube.com/watch?v=x1Jw5Y686jY
Gödel's incompleteness theorem can be stated simply as such: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle—something you have to assume but cannot prove”. Even Stephen Hawking himself, who tried to put forth a mathematical theory of everything in his book 'The Grand Design', reluctantly admitted in that book that Kurt Gödel, with his incompleteness theorem. 'halted the achievement of a unifying all-encompassing theory of everything'
"Note that despite the incontestability of Euclid's postulates in mathematics, (ref. on cite), Gödel's incompleteness theorem (1931), proves that there are limits to what can be ascertained by mathematics. Kurt Gödel (ref. on cite), halted the achievement of a unifying all-encompassing theory of everything in his theorem that: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle—something you have to assume but cannot prove”. Thus, based on the position that an equation cannot prove itself, the constructs are based on assumptions some of which will be unprovable." Cf., Stephen Hawking & Leonard Miodinow, The Grand Design (2010) @ 15-6
Gödel's incompleteness theorems were ultimately a cumulation of work that was originally started by Georg Cantor who lived from 1845 to 1918
Kurt Gödel Excerpt: Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell,[3] A. N. Whitehead,[3] and David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor. https://en.wikipedia.org/wiki/Kurt_G%C3%B6del Naming and Diagonalization, from Cantor to Gödel to Kleene - 2006 Excerpt: The first part of the paper is a historical reconstruction of the way Godel probably derived his proof from Cantor's diagonalization, through the semantic version of Richard. The incompleteness proof-including the fixed point construction-result from a natural line of thought, thereby dispelling the appearance of a "magic trick". The analysis goes on to show how Kleene's recursion theorem is obtained along the same lines. http://www.citeulike.org/group/3214/article/1001747
Cantor was trying to bring a systematic understanding of infinity into mathematics. This endeavor was very much a theological quest for Cantor. Cantor, although he pioneered some very useful tools in mathematics, useful tools which were essential to Gödel in his work on incompleteness, Cantor, none-the less, ultimately failed in his endeavor to bring a systematic understanding of infinity into mathematics. This following video gives us a very interesting glimpse into how Gödel was able to pick up the pieces where Cantor had failed, and bring his incompleteness theorems for mathematics to fruition:
BBC-Dangerous Knowledge - Part 1 (Cantor, Boltzmann, Gödel and Turing) - video https://vimeo.com/30482156 Part 2 https://vimeo.com/30641992
And although Cantor was ultimately proven wrong by Gödel in his quest to fully integrate a systematic understanding of infinity into mathematics,,,,
The God of the Mathematicians - by David P. Goldman - 2010 The religious beliefs that guided Kurt Gödel’s revolutionary ideas Excerpt: That is Cantor’s “continuum hypothesis,” which attempts to identify a first and second transfinite cardinal number. From there, he believed, all the possible orders of infinity could be counted, the same way the integers count groups of one, two, three, and so forth. He not only recognized, but was driven by, the ontological implications of this assertion: If the continuum hypothesis turned out to be true, Spinoza would be vindicated because God’s infinity could be packaged into a neat series of numbers. Cantor spent the last thirty-five years of his life in a vain effort to prove this. He died in 1918 in a mental hospital. It was Gödel and, later, Paul Cohen who demonstrated respectively that Cantor’s continuum hypothesis could be neither proved nor disproved within existing set theory. Indeed, Cantor’s hypothesis remains maddeningly undecidable.,,, God’s infinitude remains safe in heaven. Mathematicians have proven that an infinite number of transfinite numbers exist but cannot tell what they are or in what order they should be arranged. https://www.firstthings.com/article/2010/08/the-god-of-the-mathematicians
,,, none-the-less, I believe that both Cantor, and Gödel, (since they were both Christians), would be very pleased, since it is very comforting to overall Christian concerns, to see how infinity is now used, mathematically, in Quantum Mechanics and in modern day Quantum Electrodynamics,
Double Slit, Quantum-Electrodynamics, and Christian Theism - video https://www.youtube.com/watch?v=AK9kGpIxMRM
From the preceding video, it is also very interesting to note that Quantum-Electrodynamics was the first attempt at unifying relativity and quantum mechanics. Specifically, in Quantum-Electrodynamics special relativity was merged with electromagnetism to produce what is held to be the 'most precise theory of natural phenomena ever developed':
Theories of the Universe: Quantum Mechanics vs. General Relativity Excerpt: The first attempt at unifying relativity and quantum mechanics took place when special relativity was merged with electromagnetism. This created the theory of quantum electrodynamics, or QED. It is an example of what has come to be known as relativistic quantum field theory, or just quantum field theory. QED is considered by most physicists to be the most precise theory of natural phenomena ever developed. http://www.infoplease.com/cig/theories-universe/quantum-mechanics-vs-general-relativity.html
Richard Feynman was only able to unify special relativity and quantum mechanics into Quantum Electrodynamics by quote unquote “brushing infinity under the rug” with a technique called Renormalization
THE INFINITY PUZZLE: Quantum Field Theory and the Hunt for an Orderly Universe Excerpt: In quantum electrodynamics, which applies quantum mechanics to the electromagnetic field and its interactions with matter, the equations led to infinite results for the self-energy or mass of the electron. After nearly two decades of effort, this problem was solved after World War II by a procedure called renormalization, in which the infinities are rolled up into the electron’s observed mass and charge, and are thereafter conveniently ignored. Richard Feynman, who shared the 1965 Nobel Prize with Julian Schwinger and Sin-Itiro Tomonaga for this breakthrough, referred to this sleight of hand as “brushing infinity under the rug.” http://www.americanscientist.org/bookshelf/pub/tackling-infinity
bornagain77_{February 15, 2018
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asauber,
Why would I bother? What in infinity could I would I find meaningful if I read it?
You might discover why people say that some infinities are bigger than others.daveS_{February 15, 2018
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Have you read Cantor’s proof that the real numbers cannot be put into 1-1 correspondence with the natural numbers?
DaveS, Why would I bother? What in infinity could I would I find meaningful if I read it? Andrewasauber_{February 15, 2018
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asauber, Have you read Cantor's proof that the real numbers cannot be put into 1-1 correspondence with the natural numbers?daveS_{February 15, 2018
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but this is accepted as true by nearly everyone who has examined the issues seriously
Wow. A knee-jerk Appeal to Authority or Size Nearly Everyone is Pretty Big. But is it as big as a larger infinity? Guess the smart people will tell me. Andrewasauber_{February 15, 2018
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It turns out that the set of all points on a continuous line is a bigger infinity than the natural numbers;
There are always philosophical questions lurking in the background, but this is accepted as true by nearly everyone who has examined the issues seriously, regardless of their "worldview".daveS_{February 15, 2018
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a bigger infinity
Someone seriously wrote this? Andrewasauber_{February 15, 2018
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