Talk:Infinity

"The discovery of exacting and ultra-exacting cardinals represents a significant advancement in set theory and large cardinal theory."

https://www.popularmechanics.com/science/math/a63121596/exacting-cardinal-infinities/

https://www.newscientist.com/article/2459158-mathematicians-have-discovered-a-mind-blowing-new-kind-of-infinity/

https://arxiv.org/pdf/2411.11568

Juan Aguilera, a co-author of the paper from Vienna University of Technology

69.181.17.113 (talk) 02:09, 13 December 2024 (UTC)

I would suggest starting with the large cardinal page rather than here. This article should probably mention large cardinals at a high level (which currently it doesn't seem to), but not get into too much detail. --Trovatore (talk) 02:16, 13 December 2024 (UTC)

Agreed on both. - CRGreathouse (t | c) 15:47, 13 December 2024 (UTC)

I'm still trying to figure out if I think the image of the Sierpiński triangle is a good idea. I have to admit I can't think of anything better off the top of my head. It's certainly better than just an image of the symbol, and probably better than at least the current image of the reflecting mirrors, which I think is visually kind of confusing and noisy.

But in any case I copyedited the caption. I don't know what it means for something to "contain an infinite amount of itself". I tried to replace it with something meaningful and accurate; not sure if it's too wordy. --Trovatore (talk) 19:30, 16 January 2025 (UTC)

I think the Sierpiński triangle is a good image here, and I think your text is fine. Personally I'd scrub the "because of its recursive pattern" to keep it snappy but I don't mind the current version. - CRGreathouse (t | c) 02:42, 17 January 2025 (UTC)

Yeah, I considered that. With your encouragement I'll go ahead and kill it. --Trovatore (talk) 02:45, 17 January 2025 (UTC)

We could use its animated GIF or another fractal zoom GIF which I think makes the recursion a lot more self-evident if that is the way we want to go, but in my opinion having a vanishing point image like File:Ambigram tessellation Milan - concentric circles.png portrays it very well without being overly distracting. It would work great for the printable version of the article too. I'd be against using any real-world example like the last image because there are none that truly exist. --– Mullafacation {◌͜◌ talk} 19:20, 18 January 2025 (UTC)

I'm not a huge fan of animations in articles, but there's something to be said for the Sierpiński zoom. Could we make it sub in the static version for printable?

For the Milan thing, to be honest, I'm not sure what I'm looking at there.

As to whether there are any "that truly exist", that's sort of an open question. I've kind of toyed around in my head with the image at right (read the caption). I'm sure we could find a prettier one if we wanted to go that way. But the main thrust of this article seems to be mathematical rather than physical, so I kind of think probably not, but it's worth throwing out there. --Trovatore (talk) 19:45, 18 January 2025 (UTC)

If we go with the animated GIF, the printable version will be the first frame which looks the same as the image we have. And there's nothing special about the Milan image, it just looks like a tunnel with no ending. The pi image does the same thing but it is also serves as a practical mathematical example so I would prefer that. I did see in the article that it's an open question about the universe but I meant we can't show a picture that truly shows the infinity of anything in the real world. If it's an open question, it's not the strongest example we can choose. I don't know if there's any examples of images that suggest that the universe might be infinite but the image there doesn't. With pi there's no obvious reason why it's infinite while with the Sierpiński triangle you can see by looking at it how it is able to fit inside itself and go on forever. – Mullafacation {◌͜◌ talk} 21:54, 18 January 2025 (UTC)

OK, I'm on board with something like the Sierpiński animation, but I would prefer higher resolution and maybe a little pause on the first frame. Right now it kind of looks "off-balance" as it immediately descends into a non-centered part of the image. I wonder if someone has come up with an animated version of SVG? --Trovatore (talk) 22:10, 18 January 2025 (UTC)

I do not think that the Sierpiński triangle ia a good first image, because the reader must have a rather good knowledge of infinity to understand where is the infinity of this image. IMO, this image is much more pedagogical:

D.Lazard (talk) 20:17, 18 January 2025 (UTC)

I think that's too symbolic, too text-based for my liking. The Sierpiński image at least moves away from symbolic representation.

We could also consider the zero option. When it's too hard to find an appropriate image, it's worth thinking about whether an image is really needed. Shoving in images purely pro-forma doesn't appeal to me.

That said, I do think the Sierpiński triangle is better than nothing, and I also prefer it to the decimal representation of π. --Trovatore (talk) 20:41, 18 January 2025 (UTC)

I agree. I don't like putting the decimal approximation of pi. There are already too many people that have the misunderstanding that "pi is infinite." seberle (talk) 23:21, 20 January 2025 (UTC)

I noticed that infinity in this article is associated with Greek history, primarily, which documented it around 610–546 BC. Their concept was Apeiron (the infinite) as cosmic origin. However, Yajurveda documents it around 8th century BC or earlier. Their definition was in Infinity you minus or add parts, it still remains infinity.

I plan to make edits to this effect, and give equal or more weight to this concept being a Indian (both Hinduusm, and Jainism) concept.

From the vedic period the philosophical nature of infinity has been the subject of many discussions among philosophers.

And obviously add a full section on Indian vedic concept. Buddhimatta (talk) 06:28, 3 September 2025 (UTC)

So first of all, given reliable sources that have made the connection, I do think it may be reasonable to treat ancient Indian concepts to some extent. We do already have a small subsection of the History section, called "Early Indian", which could probably be profitably expanded.

That said, it seems to be the consensus that this article is to be largely focused on the notion of infinity in mathematics. We have a separate infinity (philosophy) article that might be a better home for some of this history. I don't know that I completely agree with that division but it is the current one.

You should also be aware that there is a bit of a history of editors attempting to expand the treatment of the Jain notions for what seemed likely to be nationalistic (or possibly just anti-Western) reasons. There was an editor called Jagged 85 who got community-banned in the most overwhelming way I've ever seen, more than a decade ago, for widespread misrepresentation of sources, especially on video games, but at some point s/he also turned his attention to the transfinite number article, and was pushing the Jain stuff hard. This left a fairly bad taste that has not entirely gone away. That doesn't mean the material shouldn't be covered, but it would be a good idea to find good sources (if possible, ones without an axe to grind, though this is not an absolute requirement) in advance of editing. --Trovatore (talk) 21:12, 3 September 2025 (UTC)

Thanks a ton for that thoughtful response and the historical context, @Trovatore

While I do belong to India, I am not religious. I have to acknowledge that I learned about concept of infinity early on due to my parents being religious (Jains). But as a editor, I am here to make wikipedia more trustable and factual. I wanted to focus on expanding the documented history, which from my understanding starts with the Hindu Vedas.

I have put together draft timeline that I was hoping to get a consensus on and then build the article modifications or new sections accordingly.

I also agree with you that the "mathematical concept" and the "philosophical concept" are so intertwined that it should be one article: Philosophy gives meaning to infinity, mathematics gives structure. Both are two ways of talking about the SAME concept. Kinda like "∞" is same as "Infinity".

Infinity Original Documented Timeline:

1. Yajurveda (Vedic India) Approximate Date: 8th century BC or earlier (~2800–2900 years ago from 2025) Trusted Sources: Chapter 2: Vedic and Puranic Cosmology https://www.researchgate.net/publication/2180120_Concepts_of_Space_Time_and_Consciousness_in_Ancient_India Atharva Veda (Vedic India) Approximate Date: 1000–800 BC (2800–3000 years ago from 2025) Trusted Sources: https://www.britannica.com/topic/Vedic-religion
2. Anaximander (Greek) Approximate Date: 610–546 BC (~2550 years ago from 2025) Trusted Sources: https://www.researchgate.net/publication/312595450_FROM_THE_INFINITY_APEIRON_OF_ANAXIMANDER_IN_ANCIENT_GREECE_TO_THE_THEORY_OF_INFINITE_UNIVERSES_IN_MODERN_COSMOLOGY Anaxagoras (Greek) Approximate Date: 500–428 BC (2450 years ago from 2025) Trusted Sources: https://books.google.co.in/books?hl=en&lr=&id=240MF0fzc8wC&oi=fnd&pg=PR1&ots=Hxow0A6spu&sig=k7fwH1SNAJTOb0xBAWT2XTMFPss&redir_esc=y#v=onepage&q=infinity&f=false
3. Jainism (Mahavira) Approximate Date: 6th century BC (~2600 years ago from 2025) Trusted Sources: https://pluralism.org/mahavira + https://books.google.co.in/books?redir_esc=y&id=jdjNkZoGFCgC&q=infinity#v=onepage&q=infinite&f=false + https://www.scribd.com/document/776077970/The-BIG-BOOK-of-Jain-Concepts#content=query:infinity,pageNum:48,indexOnPage:0,bestMatch:false
4. Aristotle (Greek) Approximate Date: 384–322 BC (~2350 years ago) Trusted Sources: https://plato.stanford.edu/entries/aristotle-mathematics/

Buddhimatta (talk) 15:37, 4 September 2025 (UTC)

@Buddhimatta: Please, do not use AI to reply to others. These comments may be collapsed per WP:AITALK. Leonidlednev (T, C, L) 15:50, 4 September 2025 (UTC)

Hi @Leonidlednev, Your analysis is False Positive. How does my hand written 2.5 hours of research get flagged as AI generated?? Are you using some AI tool which is actually not working. Will await your answer. Many thanks in advance. Buddhimatta (talk) 07:54, 5 September 2025 (UTC)

@Buddhimatta: Sorry for the late reply (and the FP). I originally thought that the paragraph was generated by ChatGPT due to the presence of utm_source=chatgpt.com in the source URLs. Due to that, it triggered the filter, and made me check out the comment. In the future, make sure to check the sources and remove the tracking parameters from the URLs (right-click > copy clean link) so they are not tagged as being generated by AI. Leonidlednev (T, C, L) 18:31, 7 September 2025 (UTC)

thank you Buddhimatta (talk) 06:37, 9 September 2025 (UTC)

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)

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)

...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)

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

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 , 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)

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)

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)

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)

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)

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