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May 23

Acceptability of Cantor's Transfinite numbers

Hi all, I saw the article on controversy surrounding Cantor's theory (Controversy over Cantor's theory) but am looking for a more practical answer. That is, are things like Transfinite numbers now widely accepted? If you were in college, studying in a subject area that included a topic like that, would it almost always be introduced to you as a widely accepted thing (like Darwin's theory, say) or does it depend on the professor? Best - Aza24 (talk) 00:04, 23 May 2021 (UTC)[reply]

They are fully accepted in "mainstream" mathematics, which includes everything that can be built on ZFC. There remain, however, mavericks who think the arguments for accepting ZFC as a foundation are not particularly strong. There is no common basis that could be used for demonstrating that they are wrong (or right). For much of mathematical practice, and definitely for applied mathematics, the issues are irrelevant. --Lambiam 00:26, 23 May 2021 (UTC)[reply]

Finitist mathematics (math that does not accept the axiom of infinity) does have its proponents. One prominent one on YouTube is N J Wildberger, and he explains his reasons in his video Why infinite sets don't exist. As you say, for most practical purposes these issues are irrelevant, but there is still room for disagreement among the people who specialize in foundations and philosophy, and there are many schools of thought as to which axioms should or should not be accepted. --RDBury (talk) 01:44, 23 May 2021 (UTC)[reply]

Constructivist mathematicians generally have no issue with the notion that there are infinitely many natural numbers – or, rather, that if one starts counting 0, 1, 2, ..., one will never run into some "finish" from which one cannot go further, a dead end – but the definition of cardinality used by Cantor (or classical mathematicians in general) is not meaningful to them, which undermines the whole transfinite edifice beyond

\aleph _{0}

. See Constructivism (philosophy of mathematics) § Cardinality for a (not totally enlightening) discussion. --Lambiam 10:56, 23 May 2021 (UTC)[reply]

I wouldn't say "controversies" on this topic really exist any more, and there aren't real mathematical disagreements either. There are disagreements over philosophy but that's a little bit different. E.g. asking whether transfinite numbers "exist" presupposes a level of mathematical Platonism which is a philosophical viewpoint, in which which there are various differing sub-viewpoints. The current situation is that most mathematics can be translated into a formal system called ZFC, that in turn proves theorems about sets that can be interpreted as transfinite ordinals, and nobody afaict seriously doubt that it "works" if you do it that way. Of course just because something can be done doesn't automatically make it a good idea. 2601:648:8200:970:0:0:0:752 (talk) 07:14, 24 May 2021 (UTC)[reply]

Per the above, there are really two questions here: 1) How accepted are Cantor's main notes on the difference between the infinity of counting numbers and the infinity of real numbers? and 2) How accepted are the explanations, theories, and solutions to any problems created by question #1? The answer to the first is pretty much universal acceptance. There really is NOT a one-to-one correspondence between the natural counting numbers (i.e. 1, 2, 3, 4, etc.) and the real numbers, i.e. all of the possible numbers between 0 and 1 (i.e. 0.1, 0.11, 0.111... etc). Cantor's diagonal argument really is almost brilliantly bulletproof in its application at least proving that idea. So #1 is well established. The issue comes with #2: How do you incorporate this idea of non-correspondence into existing mathematics in a coherent way? How do you generalize and formalize this? What kinds of new mathematics does this introduce? What are useful ways to think about this? We're still mostly on board with the answers to these follow on questions, which is to say there is a clear "mainstream" viewpoint within mathematics, but this viewpoint is not universal, and there are still good-faith disagreements over this part. This is where this becomes a "philosophical" rather that a "mathematical" argument. --Jayron 32 14:02, 25 May 2021 (UTC)[reply]
I do have to say, in addition, that these philosophical arguments are also not just trivial or academic exercises. There are all sorts of real applications that depend on these mathematical concepts, including computer science (i.e. the halting problem), cryptography (i.e. P versus NP problem), quantum mechanics issues, etc. that all come down to the ways we resolve these set theory problems. The Russell paradox problem keeps showing up in logic systems where self-reference is allowed, and avoiding self-reference is itself fraught with complexities; Russell's best attempt a creating a system of logic robust enough to avoid self-reference issues introduced by naive set theory is Principia Mathematica. ZFC solves the problem with the axiom of choice. --Jayron 32 14:11, 25 May 2021 (UTC)[reply]
I don't really see how the axiom of choice "solves the problem". I would put it rather that the problem is solved by conceiving of sets as objects contained in the cumulative hierarchy edit: I followed the link and that's actually not quite the article I want — see von Neumann universe instead, in which the axiom of choice is "obviously" true (scare quotes because it's obvious to me, but not to everyone). It's actually the axiom of foundation (which our article unfortunately calls "regularity", a silly name for it) that encapsulates the claim that all sets are contained in the cumulative hierarchy.

But I wouldn't say it's the axiom of foundation that "solves the problem" either; you can't get rid of a contradiction by adding more assumptions. It's rather a matter of getting the informal description correct; then you can see that the axioms that lead to the contradiction just aren't true. --Trovatore (talk) 17:22, 25 May 2021 (UTC)[reply]

If follows from the Church–Turing–Deutsch principle "The principle states that a universal computing device can simulate every physical process" that all of mathematics that mathematicians can ever invent, can always be reduced to discrete mathematics. This is because the very act of doing mathematics is a physical process which can be simulated with arbitrary precision using a large enough computer. Count Iblis (talk) 06:04, 27 May 2021 (UTC)[reply]

That's like saying the Chinese Room can speak Chinese. True in some sense, but not in any useful sense. --Trovatore (talk) 17:26, 27 May 2021 (UTC)[reply]

The principle is not helpful in seeking a resolution to the seemingly elementary issue whether the law of excluded middle is trustworthy when applied to infinite sets.^[1] --Lambiam 19:06, 27 May 2021 (UTC)[reply]