Talk:Infinity

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This article was selected as the article for improvement on 20 May 2013 for a period of one week.

Original comment by Victor Kosko, awkwardly put in the heading instead of the body: You removed all references to hyperreals, presumably from set theory prejudice. NOT redundant. But if you want to include ALL types of infinity: Cardinals, ordinals, infinitesimal of infinitesimal calculous, measure theory, the infinities of limits of sums products and integrals, and axiomatic idealization of arbitrarily large finite number which is by far the most common

removed hypperreals Victor Kosko (talk) 03:42, 20 January 2026 (UTC)[reply]

For what it's worth I agree that this article should mention the infinite objects considered in nonstandard analysis. --Trovatore (talk) 04:12, 20 January 2026 (UTC)[reply]

...aaand, in fact, it does. I wrote the above comment without checking the article. It treats hyperreals in the appropriate section. The mention that ALittleClass removed was out of place in context, coming after a sentence in the set-theory section talking about mathematics accepting actual infinity in a way consonant with Cantor's treatment. --Trovatore (talk) 00:44, 21 January 2026 (UTC)[reply]

! Victor Kosko (talk) 01:55, 21 January 2026 (UTC)[reply]

Originally after "The mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them" the article cited the fact that Wiles' Proof of Fermat's Last Theorem relies on the existence of Grothendieck universes which I have removed. At best, this is controversial, and at worst it is plain wrong (see the experts squabbling here [1], including one of the coauthors of the proof of the Modularity Theorem).

Therefore, this example should be replaced with something equally interesting, but I cannot think of anything suitable off the top of my head. Doable7366 (talk) 00:37, 8 March 2026 (UTC)[reply]

It's plausible that Wiles' proof, as written, uses inaccessible cardinals. It does seem to be the general opinion that they can be removed, and McLarty had a whole program on reducing the level of math required (and I think he got pretty far, in principle, though I think he was just hitting the obvious high points rather than doing a deep dive to ensure the whole proof was clean). - CRGreathouse (t | c) 02:33, 9 March 2026 (UTC)[reply]

In any case I agree that finding another example would be nice, though I don't know one off the top of my head. Szemeredi's lemma requires at least a tower of exponentials to prove, so that's something. - CRGreathouse (t | c) 03:16, 9 March 2026 (UTC)[reply]

Well, it certainly isn't "plain wrong". As far as I can tell from the discussion, it's clear that the proof as written uses assumptions that entail inaccessibles. What's a little controversial is whether this use is essential, and the smart money seems to be on "almost definitely not", but that is a different thing; a non-essential use that no one has shown how to remove is still a use. (Actually even if we did know how to remove it, it would still be a use.) --Trovatore (talk) 23:42, 9 March 2026 (UTC)[reply]

As far as I can tell from the discussion, it's fairly uncontroversial that Wiles himself does not actually use Grothendieck universes, and they're somewhere buried in a chain of references. According to said users, he cites SGA/EGA results which need Grothendieck universes to state in the manner they are stated by Grothendieck, but Wiles would only need a much weaker result and therefore it would be easy to remove—cohomological number theorists just have better ways to spend their time.

To draw an analogy, if one was proving a result in (finite) combinatorics, they might cite a result that cites a result that A is equinumerous to B if and only if there are surjections from A to B and from B to A. Of course in the original proof of this fact, the axiom of choice may have been used (assuming it was just the general proof for any two sets), but our hypothetical person, doing purely finite combinatorics, didn't actually need the axiom of choice. Did they really use it then? Technically, but choice-style arguments never entered their thought process, and the result doesn't really need choice, so it's misleading to claim they really did use choice. Mutatis mutandis with Wiles and Grothendieck universes and sheafification at the crystalline site or whatever.

This brings me to my next reason to delete it. Even supposing that I am wrong and it is a logical necessity which takes some nontrivial work to unravel, according to BCnrd nobody in cohomological algebra is thinking in terms of Grothendieck universes, so claiming that Wiles is using them is only technically true.

Anyways, supposing we should include it, what should our reference be? McIntyre's paper has been contested by some of the foremost experts in Fermat's last theorem, so surely it's not great to cite a paper with an undefeated defeater like this.

P.S. I don't think your reply was necessarily meant to insinuate we should restore it. I just like the sound of my own keyboard, so to speak. Doable7366 (talk) 02:36, 10 March 2026 (UTC)[reply]

OK, that sounds at least plausible. Thanks for clarifying. --Trovatore (talk) 07:23, 10 March 2026 (UTC)[reply]

In the first sentence, it says "Infinity is something which is boundless, limitless, endless." Shouldn't it be "Infinity is something which is boundless, limitless, and/or endless?" DanyMations (talk) 18:03, 4 April 2026 (UTC)[reply]

The redirect InFINity has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2026 April 14 § InFINity until a consensus is reached. consarn _{(talck) (contirbuton s)} 20:51, 14 April 2026 (UTC)[reply]

The redirect Infinite (synonyms) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2026 April 15 § Infinite (synonyms) until a consensus is reached. consarn _{(talck) (contirbuton s)} 12:14, 15 April 2026 (UTC)[reply]