Talk:Infinity/Archive 4

@Deacon Vorbis:

how any of these links are an actual problem to the any reader...

Why it is there is an article titled concept since it is such an ordinary word, if your understanding were true, no-one would need to read the article, since the topic is such an ordinary fact of life 23h112e (talk) 21:04, 26 November 2017 (UTC)

Why your opinion should supercede my own in any case, while you state policy, although how, since there is the possibility of error on your account, as indicated by ther confusion on your part between ordinary and everyday, and the non-absolute description in policy of behaviours with respect to under and overlinking, a factor to which I refer to due to you so helpfully including mention in your editorial summary. 23h112e (talk) 21:13, 26 November 2017 (UTC) 23h112e (talk) 21:14, 26 November 2017 (UTC)

Side note: you don't need to ping me every time you write here; I'm watching this talk page. "Ordinary" and "everyday" are synonyms here. You seem to have difficulty with English; I don't mean this in a disparaging way at all; I'd be pretty hard-pressed to communicate in any language other than English, myself. But your confusion over "everyday" versus "ordinary" is a problem here. It's also difficult to follow what you're trying to say. While I think I get the overall gist of what you're saying, it's hard to be sure exactly what you mean, because the English you're using is so awkward (I've had the same problem with many other comments you've written, too). One possibility is to try editing the Wikipedia of whatever your native language is; it could probably use some help too, and you may be more successful when you need to communicate with others.

But back to the overlinking, "abstract" and "concept" are ordinary, everyday words. It's especially important in the opening sentence not to link to things that aren't really directly related because such links are just a distraction. –Deacon Vorbis (carbon • videos) 23:22, 26 November 2017 (UTC)

for example:

any number, the largest number is... (there isn't one) obviously, there is no largest number - unless it is a counted number of actual things, anything obviously.

To re-iterate, the largest number actually of any concrete thing is perhaps the number of atoms in the observable universe (which is uncountable in any case, since no-one knows the entire universe), the number of stars in the observable universe - which is presumably a number which might be in the future known; or might already be known - is a number; +1 of this second number does not equal infinity.

Anything which is an actual observable thing, which has any relationship to infinity since infinity by definition - is never reached, is not graspable, is not an actual number, (to re-iterate again) since any number stated is therefore having +1 as a larger number (or +0.00000...n...>1).

Numbers refer to actual things, for those things to have number applied to them, to be counted, even if it is a non-known number, for example - atoms in the universe, there is still the +1 factor which therefore denies the infinity to the number.

"without bound" is obviously the actual correct definition, because, if it were possible to imagine any situation where infinity might occur in the universe, it is simply a matter of lack of knowledge of the actual number of the things, which allows the incorrect presumption of infinitesimal. Every thing in the universe has number, without bound suggest the theoretical travelling in any direction from earth, to the boundaries of the universe > , and the thing which is beyond - this is the true and only actual referent to infinity, infinity is ipso facto this definition and this does not include the "larger than any number" definition.

23h112e (talk) 22:02, 19 November 2017 (UTC)

@Deacon Vorbis: 23h112e (talk) 22:06, 19 November 2017 (UTC)

https://en.wikipedia.org/w/index.php?title=Infinity&action=history:

• 20:55, 19 November 2017
• 20:52, 19 November 2017‎
• 20:02, 19 November 2017
• 18:49, 19 November 2017‎

23h112e (talk) 22:11, 19 November 2017 (UTC)

Your comment is kind of a jumbled mess; I'm not really sure how to respond to it, but I'll make a few points anyway. In terms of numbers, the ordinal number ω is in fact larger than any integer, so it depends on what number systems we're talking about. This is the lead of the article – an article that's about a vague, abstract concept. A certain amount of latitude is understandable in trying to discuss it. Moreover, the article lead isn't (just) for defining the subject of the article; it's for summarizing the key points in the rest of the article. Moremoreover, things (infinity included) don't always have a single description or definition; they can have many. And as for "numbers refer to actual things" – they might, but they might also refer to fake things, or to no things at all. I'm really at a loss as to your objection here.

Side note, I also don't know why you posted a bare URL of the article history along with some dates. --Deacon Vorbis (talk) 22:27, 19 November 2017 (UTC)

Well, 23h112e does make at least one useful point here. We're deliberately vague on whether infinity is a "number" (see lots of discussions in the talk history). But if infinity is in fact a number, then it can't be larger than "any number", because it isn't larger than itself.

I will change this bit to "larger than any natural number", as a stopgap. That is actually one of the definitions of infinite quantity. I'm not trying to shut off discussion; there could be better solutions. --Trovatore (talk) 23:46, 19 November 2017 (UTC)

@Deacon Vorbis: @Trovatore:

try imagine https://en.wikipedia.org/wiki/%CA%BBOumuamua as it appears in the moving image here, if it were to continue, then it would eventually cease due to lack of energy, or impact, although to then project a course in the direction shown, since the universe cannot have a boundary, it would (in the imagined situation; circumstances) continue forever in the direction "without bound" suggest the theoretical travelling in any direction from earth, to the boundaries of the universe as shown at 22:02, 19 November 2017 (UTC) 23h112e (talk) 18:24, 21 November 2017 (UTC)

I utterly fail to see your point. What does all that have to do with the price of tea in China? --Deacon Vorbis (talk) 01:28, 22 November 2017 (UTC)

@Deacon Vorbis:

actually the price of tea in China currently (a cake of two-year-old Ye Sheng Gucha tea costs 260 yuan (about £18) is about a million miles away, or more, from the focus of the current discussion - tell me where your mind is heading to, which direction of thought were you intending to take me in ? (and I don't eat cake in any case, by the way) 23h112e (talk) 17:13, 23 November 2017 (UTC)

Extraordinary 23h112e (talk) 17:17, 23 November 2017 (UTC)

Please indent your messages when replying to someone; see Help:Talk for more detail. Also, pinging doesn't work unless you sign your post when you use it. As for the tea thing; see wikt:price of tea in China for a definition. The point was that nothing you said in your previous post seemed to have any relevance to anything about the article here. --Deacon Vorbis (talk) 17:31, 23 November 2017 (UTC)

@Deacon Vorbis: c.f. (ignoring the links to youtube) - https://en.wikipedia.org/wiki/User:23h112e/sandbox#26th 23h112e (talk) 21:06, 26 November 2017 (UTC) hoping this is sufficiently elucidative for you, considering your apparent doubting of my actual abilities to grasp something, although sitting here I'm sure I see and understand something you haven't quite completely grasped about me, to allow you to be so free as to pass comment on anything about me with certainty 23h112e (talk) 21:17, 26 November 2017 (UTC)

I'm thinking because i'm actually correctly describing uncountable, and you have found me to do so, you therefore have concluded I am defined as an individual also uncountable, which therefore some-how empowers you to negate my contributions by your own subjective logic. 23h112e (talk) 21:20, 26 November 2017 (UTC)

No, your link isn't specific, and it doesn't help me understand your point. I still don't understand what you're trying to say. Nothing about the astronomical body had anything to do with the article or what you had tried to remove from it. –Deacon Vorbis (carbon • videos) 23:26, 26 November 2017 (UTC)

@Deacon Vorbis: hello from here Deacon (not philosophy instead) - "The layman often perceives it as a kind of "number" larger than all numbers..." Princeton University Press, 1991 Eli Maor is a teacher of the history of mathematics who has successfully popularized his subject with the general public

original search page, criteria is "ancient Chinese infinity philosophy" text - p.2 I'd like to know your opinion on the statement by Eli Maor (additionally considering is a teacher of the history of mathematics (doesn't mention professor)) 23h112e (talk) 16:54, 19 December 2017 (UTC)

Yes, it's a statement (which loses some meaning when taken out of context). What about it are you asking my opinion of exactly? –Deacon Vorbis (carbon • videos) 17:10, 19 December 2017 (UTC)

In hindi, we call it anant not ananta. And it is the problem that in English, they include "a" at the end. अनंत is the spelling and we read it as anant.
117.207.26.105 (talk) 20:11, 14 May 2015 (UTC)

I think "ananta" is appropriate to use in english as per rules of grammer Navjot1200 (talk) 16:19, 24 January 2018 (UTC)

I think it is a concept rather than a Number as considered in Mathematics.... Navjot1200 (talk) 16:18, 24 January 2018 (UTC) Navjot1200 (talk) 16:18, 24 January 2018 (UTC)

The very first sentence of the article already calls it a concept. Later in the article, there's some discussion on various number systems that involve infinite quantities. There's also Infinity (philosophy) for less mathematical views on the subject. –Deacon Vorbis (carbon • videos) 16:25, 24 January 2018 (UTC)

I think an article on the Christian theological concept of infinity should be added. As the Catholic Encyclopedia states: "When we say that God is infinite, we mean that He is unlimited in every kind of perfection or that every conceivable perfection belongs to Him in the highest conceivable way. In a different sense we sometimes speak, for instance, of infinite time or space, meaning thereby time of such indefinite duration or space of such indefinite extension that we cannot assign any fixed limit to one or the other. Care should be taken not to confound these two essentially different meanings of the term." — Preceding unsigned comment added by 24.191.87.42 (talk) 00:19, 21 March 2012 (UTC)

• I'd be happy just to have some brainiac prove that infinity is more than a mathematical concept and has real-world application. In a world of quantum-mechanics, nothing can be measured to infinite degrees of certainty. Religious faith goes beyond what science can prove. I'm just being a realist. — Preceding unsigned comment added by 2600:6C48:7006:200:D84D:5A80:173:901D (talk) 03:38, 17 March 2018 (UTC)

I could argue that most - if not all - aspects of Infinity in the area of Physical Sciences (Physics) are purely mathematical in basis. Infinity is a mathematical concept intended to represent very large numbers that are (at least thus far) immeasurable and cannot be determined to be finite. Pure physics relies on observations to confirm its theories, and yet all measurements contain experimental error and it is not possible to observe the entirety of infinity. Lack of citation on Physics-based applications of infinity only support my position. I invite others to add to this, or update the article accordingly. I'd prefer to remove reference to Physics from discussion of Infinity, but I think others need to contribute. --68.188.183.91 (talk) 02:30, 20 February 2015 (UTC)

You are making a philosophical argument here; it's not directly relevant to what should appear in the article. No doubt there are reliable sources that have made similar arguments, and those can certainly be cited. However, it is also true, for better or for worse, that the notion of infinity has been and continues to be used in physics, and we are not going to avoid mentioning that just because you don't think physics should use it. --Trovatore (talk) 08:14, 20 February 2015 (UTC)

The article "infinity" should not be purely a mathematical concept. There are great philisophical concepts. Consider death, for sample. You can't measure infinity, or any physical constant to infinite degrees of certainty. Throw in quantum mechanics - there is no certainty. You can't talk about infinity without uncertainty.--2600:6C48:7006:200:D84D:5A80:173:901D (talk) 03:28, 17 March 2018 (UTC)--2600:6C48:7006:200:D84D:5A80:173:901D (talk) 03:28, 17 March 2018 (UTC)

See Infinity (philosophy). Paul August ☎ 12:23, 17 March 2018 (UTC)

@Joel B. Lewis: Per MOS:FORMULA that you cited, "[...] from non-LaTeX to LaTeX without a clear improvement," I believe there is a clear improvement and keeps consistency in the article (WP:ARTCON); LaTeX would be preferred over {{math}} as the symbols are clearer, in my opinion; in addition, "English Wikipedia currently has no consensus about preferred formatting," and "Large scale formatting changes to an article or group of articles are likely to be controversial," therefore I do not know how you can say it is discouraged when it explicitly states it is "likely to be controversial." {{u|waddie96}} {talk} 12:21, 30 October 2018 (UTC)

Your claim that one is clearly better is immediately contradicted by the fact that there is no consensus. The fact that you were reverted means that it was controversial. It is definitely not an improvement to use LaTeX for individual small inline formulas, given current viewing options. (Maybe at some point it will be -- that will be nice -- but it isn't now.) --JBL (talk) 13:31, 30 October 2018 (UTC)

@Joel B. Lewis: "It is definitely not an improvement" is your opinion. Contradicting your first statement that there is no consensus. Do you see the issue here? Please go through the article then and change all inline formulas to {{math}} instead of <math> to maintain consistency per your opinion. There needs to be either or Joel. {{u|waddie96}} {talk} 18:23, 30 October 2018 (UTC)

Joel B. Lewis is certainly not alone with this view. There are several experienced math editors (but certainly not all) who would agree with him, myself included. In-line LaTex, with its alignment, size and intensity issues can and does at times make the visual presentation awkward. There are times when its use can't be helped, but these are relatively rare. Wholesale LaTexification is really not a good option at this point in time, and certainly not without some consensus on the talk page for it.--Bill Cherowitzo (talk) 18:59, 30 October 2018 (UTC)

In the "Cardinality of the continuum" section, the diagram displays the first three steps of a fractal to generate a space filling curve.

I think that the first step is wrong: it seems that the horizontal lines shouldn't be right at the top and bottom of the image, otherwise in the next steps the lines would overwrite each other. Instead, the horizontal lines should only be near the top and near the bottom.— Preceding unsigned comment added by Richierocks (talk • contribs) 08:07, 15 March 2015 (UTC)

This is a terrible phrase in the article. The "concept" has already been explained as a purely philosophical item. What is it supposed to mean that mathematics uses a philosophical concept? At best it explains nothing, except it creates confusion. The truth is that the infinity symbol ∞ is used in mathematics to abbreviate expressions involving limits, typically in summations and integrals. There is no "concept" involved, since you could freely choose to replace the symbol by the more tedious original limit expressions without changing the meaning. The symbol is also used in situations, in topology, elliptic curves, and real and complex analysis, etc., when it is practical or necessary to add an additional element to an existing structure to obtain a larger structure with desired properties. The "extra" element is traditionally named using the symbol ∞. There is no concept whatsoever involved in this choice. The statement "mathematics uses a concept of infinity" is a basic misunderstanding. — Preceding unsigned comment added by 90.185.71.194 (talk) 20:14, 17 February 2019 (UTC)

The substance of what you've written supports, rather than refutes, the (true) statement that the concept of infinity is used in mathematics in various different ways. Though of course your list of places where the concept is used is far from complete, and your interpretations of the various uses of the concept are idiosyncratic at best. --JBL (talk) 20:49, 17 February 2019 (UTC)

The concept of infinity was, until the end of 19th century, a purely philosophical concept. Now, it is also a mathematical concept, which is rather different from the philosophical one; for example, in mathematics, there are infinite sets of different size, and the possible sizes have been quantified and studied (see Transfinite number). So mathematics use a concept of infinity that is not exactly the philosophical concept. I have edited the lead of the article for clarifying that. D.Lazard (talk) 21:11, 17 February 2019 (UTC)

I have some concerns about the new wording, in particular the claim that In modern mathematics the concept of infinity has been formalized. For one thing, I think it puts too much emphasis on formalization. Also the reference to "the" concept of infinity sounds as though there is only one, when in fact there are different sorts (at least three: transfinite cardinals, transfinite ordinals, extended real and complex numbers, and a possible fourth with infinities from nonstandard models). (This would be less of a problem if we weren't talking about formalization.)

I am also not sure I agree with the rationale expressed by D.Lazard above. I would agree that the modern mathematical concept(s) of infinity is (are) rather different from the historical philosophical concept, which I suppose means more or less what Aquinas thought. But Cantor arguably initiated a new philosophical concept, which continues to this day. --Trovatore (talk) 21:52, 17 February 2019 (UTC)

One may argue infinitely about what is philosophy and what is not. This been said, there was clearly a need to clarify in the lead that, in mathematics, infinity has been made more than a concept, and is rather different from the intuition that anyone can have. I think that Trovatore and I can agree on that. The difficulty is to find a formulation (in one or two sentences as needed in the lead) that is not controversial. I think that my version is better than the previous one, but it is only a first draft that needs work for reaching a consensus. Specifically, I have hesitated between "the concept of infinity" and "a concept of infinity", and also between the words "formalized", "refined", "expanded", ..., all expressing a part of what Cantor and his followers have made to the concept. In any case, a better formulation than mine would be welcome. D.Lazard (talk) 09:38, 18 February 2019 (UTC)

I like "refined" a lot. --JBL (talk) 12:07, 18 February 2019 (UTC)

I have completely rewritten the part of the lead devoted to mathematics. IMO, this is simpler and much clearer for a large audience. I guess that discussing of further improvements would be easier with this version than with the previous ones.

In particular, the axiom of infinity must appear in this article, but I am not sure whether this should be in the lead or in the body. D.Lazard (talk) 15:10, 18 February 2019 (UTC)

I like it. I would not put any particular axiom in the lead, as it strikes me as too specific to a particular axiomatization. --Trovatore (talk) 02:11, 19 February 2019 (UTC)

I have posted this under Infinity in a section I created and titled "uses in african-american theory". It got deleted for some reason and I would really like to reach out and ask for help from the community. I hope I get pointers about why it was deleted and what I can change in order to repost it and keep it there. Thank you so much for all the efforts and a the help.

" the indeterminacy of infinity in Math has been extrapolated as a philosophical and critical signifier in African-American Studies, in order to situate racism and antiblackness as determinate values of Western thought that need the indeterminacy of blackness to be defined and sustainted. Denise Ferraira da Silva, Professor and Director of the Social Justice Institute at the University of British Columbia, theorizes on the value of blackness rendered obsolete. In her article entitled “1 (life) ÷ 0 (blackness) = ∞ − ∞ or ∞ / ∞: On Matter Beyond the Equation of Value”, da Silva uses numbers and equations towards her argument that blackness exists without value and without form. Here, the mathematical use of infinity is conceptualized as a discourse of “refusal to contain blackness in the dialectal form” . Equating blackness with infinity signifiers an impossible/indeterminate value that is not bound by categories and premises of modern thought." Ktf87 (talk) 18:32, 23 March 2019 (UTC)

@Ktf87: The reasons were given in edit summaries; you can click the "View History" tab at the top of the page to see the summaries that were used along with the various edits that removed the material you've been trying to add. Since there's already a discussion at Talk:Infinity (philosophy) (which would at least be a slightly better place for this sort of material) about the same topic, I'll keep any further discussion over there rather than here – I just wanted to draw your attention to the reason(s) used for removal, since you seemed unclear. –Deacon Vorbis (carbon • videos) 20:23, 23 March 2019 (UTC)

It is pobably not Wikipedia's job to correct this. I'm guessing that primary mathematics publications must correct this error first, which I suspect will take some time.

The wikipedia definition confuses the terms unbounded (without any bound) and infinte (beyond counting). Mathematical inconsistency results from a lack of clear definitions of the terms finite, unbounded and infinite. Finite is unbounded which means that it can tend to infinity. In analysis we consider what would happen in the limit (i.e. if we could allow finite to become infinite). In algebra allowing unbounded to be infinite has always been illegal (prior to Cantor which I reject). Unbounded is best thought of as the never-ending journey from finite to infinite, it is not the same as infinite because the journey never gets there (except through analytic reasoning). Paraxoxes vanish once this is properly understood. See ^[1] and ^[2]. Epdarnell (talk) 12:56, 29 August 2019 (UTC)

It is pobably not Wikipedia's job to correct this. Indeed: see our policy WP:NOR. --JBL (talk) 13:12, 29 August 2019 (UTC)

Some reference to God (because e.g. Allah or Bible God is only solely considered Almighty) should be added to the article. Only infinity could make almightiness possible (e.g. drawing out energy to the infinite extent in order to have for the entity in question, God in this case, the ability to do whatever it/He wants).

Currently there is about 3.42 million (or 3.3m when archive.is performs search) results on "almightiness of God infinity" Google searh, so proper reliable sources can be found for sure. --5.43.99.155 (talk) 04:35, 26 September 2019 (UTC)

While WP can certainly not assert any of your points in its own voice, I actually do think there is a decent case to mention theological or otherwise spiritual aspects of the notion of infinity, and indeed they are mentioned at Absolute Infinite and at infinity (philosophy). The division of labor between infinity and infinity (philosophy) is a little unconvincing, and I wouldn't mind seeing them merged if it could be done well. As a side note, it's more than a bit odd that none of these articles mentions Aquinas. --Trovatore (talk) 05:20, 26 September 2019 (UTC)

While "very large infinite sets" can be meaningful, it's also surprising to the naive reader, and is not really explained in situ. For that matter it might be confusing to the expert reader as well, who might assume that this means that the proof uses large cardinals, which (I'm not certain but) I think is not the case.

It seems likely that it probably means it uses the set of all sets of reals, or something smaller than that. That's a "very large infinite set" in some sense, but not in all senses. We should (i) figure out what the real situation is, and then (ii) figure out how to express it in accurate but understandable language. --Trovatore (talk) 02:55, 8 October 2019 (UTC)

@Trovatore: Real quick, just FYI, I also just posted a thread over at WT:WPM about this as well. –Deacon Vorbis (carbon • videos) 02:57, 8 October 2019 (UTC)

Looking at the history, it looks like the statement was added by D.Lazard in this edit. Maybe he'll want to weigh in. –Deacon Vorbis (carbon • videos) 03:05, 8 October 2019 (UTC)

@Trovatore: I mentioned it in the WPM discussion, but the proof uses Grothendieck universes so it is about the existence of certain large cardinals in ZFC. — MarkH21 (talk) 04:55, 8 October 2019 (UTC)

Hmm ... Grothendieck universes 'aven't got much large cardinals in 'em, but still, it's more than I was expecting. That's the kind of thing that's often eliminable by a minor reworking of the proof, but of course I don't know the proof, so it's hard to say in the specific case.

Not really sure where to go from here. "Very large infinite sets" is still surprising to the naive reader and possibly misleading to the expert reader (who might expect it to mean more than piddly little inaccessibles). But no good rewording immediately leaps to mind. We certainly don't want to get into Grothendieck universes (or large cardinals, for that matter) in the second paragraph of the general article on infinity.

I see that someone has linked the Grothendieck-universe article, which is something I guess, though in general I dislike hiding information in piped links. In principle links should be for convenience, not for information that belongs in the article (they don't appear in printed versions, for example). --Trovatore (talk) 05:39, 8 October 2019 (UTC)

Yes, I just added the reference and the piped link because it was immediately better than the previous state. I don't have any strong feeling for the current state though, so feel free to tweak or remove. — MarkH21 (talk) 06:03, 8 October 2019 (UTC)

Many thanks for these edits. I have added in the preceding sentence a phrase about the "size" of infinite sets, which may also help to understand "very large". Feel free to remove it, if you think it is too much detailed. Nevertheless, I think important to insist on aspects of mathematical infinity that are paradoxal for non-mathematicians. D.Lazard (talk) 09:04, 8 October 2019 (UTC)

Yep, thanks to all for taking a look at this one. –Deacon Vorbis (carbon • videos) 14:29, 8 October 2019 (UTC)

The qualifier "very large" is controversial, as this section abundantly illustrates. In context, it is also totally unnecessary. I am deleting these two words and the associated link. The reader does not need to understand what they mean. Peter Brown (talk) 19:15, 9 October 2019 (UTC)

I'm fairly ambivalent about this note being here at all, but saying it's due simply to the existence of infinite sets makes it plainly incorrect, so I've reverted the removal of "very large" once again. It also doesn't appear to be "controversial" so much as just needing explanation/context. –Deacon Vorbis (carbon • videos) 19:29, 9 October 2019 (UTC)

I agree with Deacon Vorbis in principle - it’s kind of meaningless to say that the proof depends on the existence of infinite sets. Most of mathematics does. Pedantic note: ... uses the existence of infinite sets is correct in the literal sense though... it depends on the infinitude of the natural numbers for instance. That’s a pretty meaningless example though as it’s nothing even remotely particular to FLT. — MarkH21 (talk) 19:41, 9 October 2019 (UTC)

(edit conflict) Nobody at all has suggested that "it's due simply to the existence of infinite sets" (your wording). It is necessary to claim only that the proof uses the existence of infinite sets, my preferred wording, which is true. Specifically, the proof uses the existence of Grothendieck sets, but that's a refinement that the user need not be troubled with. In such an elementary presentation, even linking to Grothendieck universe is probably overkill, though it is appropriate to get into the matter in sourcing the statement. What's relevant, here, is only that a proof in number theory uses some kind of set that is infinite. Peter Brown (talk) 20:42, 9 October 2019 (UTC)

The topic of the paragraph is "The mathematical concept of infinity is used everywhere in mathematics… ". Further discussion is necessary only because some readers are likely to think, "Hey, that can't be true of number theory." To fostall that reaction, the pargraph needs to add, "Yes, it's even true of some proofs in number theory, including a proof of Fermat's Last Theorem, which you may have heard of. Here's a relevant source." There is no need to go into any detail as the the kind of infinite entity involved. Peter Brown (talk) 20:33, 9 October 2019 (UTC)

Just on the point of number theory in general, I don’t think that’s very likely, nor is the fact that FLT uses some infinite set all that surprising. The very statement is about the integers which is an clearly infinite set. Even the most basic result, that there are infinitely many primes, is about infinitude and due to Euclid 2300 years ago. — MarkH21 (talk) 21:00, 9 October 2019 (UTC)

D.Lazard introduced into the article the idea that "number theory … may seem to have nothing to do with it [infinity]." I concur. It may seem so to some readers, and further discussion is needed to dissuade them. Peter Brown (talk) 22:40, 9 October 2019 (UTC)

IMO, it is important to insist that, in modern mathematics, infinite sets are manipulated as actual objects not as the result of a unbounded process. The concept of actual infinity is so common for mathematicians that many forget that it is paradoxal (or even ignored) for non-mathematicians, including many philosophers. A witness of this ignorance is the body of this article itself: Cantor is not even cited in the history section; Actual infinity is linked only for saying that ancient Greeks did not accepted this concept; etc. It is for starting to correct this misrepresentation of the mathematical concept that I have edited the lead. A deeper edit of the whole article would be needed, but it would need much more work.

It seems from the above discussion that the aim of my edit may be unclear for some editors. For clarifying this, I have added a phrase about actual infinity and its manipulation. I hope that this will make clear why the use of Grothendieck universes (actual infinity) in the proof of Fermat's Last Theorem is of a different nature from Euclid's proof of the infinity of the sequence of primes (potential infinity). I am convinced that the lead must make clear that actual infinity is not only commonly used, but is fundamental in modern mathematics. It is possible that my edit could be improved, but in any case, per WP:NPOV, the importance of actual infinity must not be minimized. D.Lazard (talk) 08:56, 10 October 2019 (UTC)

@D.Lazard: I think that it is an important concept and should be mentioned. I had removed it earlier because the actual link was a WP:SURPRISE and the topic is not really covered in the body. I agree that it's important enough to include in the lead at some point, although with clearer link context and perhaps after it is mentioned more thoroughly in the article. — MarkH21 (talk) 09:30, 10 October 2019 (UTC)

Unless someone can defend the mention of Zeno in this article, I propose to delete the associated passages.

The lead mentions Zeno as one who speculated about the nature of the infinite. No documented speculations on the matter, however, are attributed to Zeno. His paradoxes are best understood in the context of the Eleatic rejection of the ideas of motion and change, not as positive advances over the philosophy of Parmenides, the founder of the Eleatic school, whose known views were quite unconcerned with the infinite.

Since the 17th century, consideration of the infinite has provided tools to resolve the paradoxes. This explains the recent association of Zeno with infinity, but it does not turn the paradoxes into speculations on the concept.

Similarly, the section § Early Greek attributes "attestable accounts of mathematical infinity" to Zeno. There are no such accounts in the known works of Zeno.

It is noted, correctly, that Aristotle called Zeno the inventor of dialectic. This fact has nothing to do with infinity and does not belong in the article.

Peter Brown (talk) 16:37, 10 October 2019 (UTC)

On the principle, I agree with you. However, for many people, Zeno's paradoxes are associated with infinity, or, more exactly with the fact that an infinite series may have a finite sum. Therefore, I suggest replacing the passages that you will suppress by a section that expands your sentence "Since the 17th century, consideration of the infinite has provided tools to resolve Zeno's paradoxes". This has to do with infinity, and thus belongs to this article. D.Lazard (talk) 18:02, 10 October 2019 (UTC)

Challenge accepted! However, since I have made no use of my mathematics and philosophy degrees in the last 30 years and I do not currently have access to an academic library, it will take me a bit to get up to speed on this. I will bear no hard feelings if some reader of this section makes the changes before I'm ready. Peter Brown (talk) 00:48, 11 October 2019 (UTC)

See Talk:Infinity (disambiguation)#Description of Infinity. D.Lazard (talk) 16:28, 27 October 2019 (UTC)

There is not really any such thing as infinity. It is therefore misleading to use "Infinity" as the name of an article. A truly encylopediac article would not start out with the qualification "In common usage…". Rather, it would characterize whatever the article title denotes or at least prepare the reader for such a characterization. When, as here, there is nothing denoted—not even something fictional—such a characterization is impossible and therefore has not been provided.

A widely held view, which I do not dispute, is that some things are infinite, the set of real numbers for example. Whether the universe is infinite in extent is certainly a coherent issue. The quality of being infinite, however, is infinitude, not infinity. The article perhaps should be renamed "Infinitude". This would need some reworking of the lead section and, of course, a redirect from Infinity to Infinitude.

To be sure, the word "infinity" is used meaningfully. Here are over one hundred samples. All such uses, though, are shorthand and do not imply the existence of something called "infinity".

When a variable is said to "approach infinity", "tend to infinity", etc., what is usually being described is the state of a system or the value of a dependent variable that is approached as a limit as an independent variable increases without bound. Some writers purport to describe the state of a system when the independent variable is "at infinity"; these really characterize a state that never actually obtains but which is approached as a limit as the variable increases without bound.

Another common use of the term is in the phrase "an infinity of". That phrase is equivalent to "infinitely many" and does not imply the existence of something called an infinity. Grammatically, the expression is plural, despite the singular article; one says "There are an infinity of natural numbers", not, "There is an infinity of natural numbers", at least not usually. Of course, the set of natural numbers does have a cardinality, ℵ₀, but that's another matter.

Peter Brown (talk) 05:08, 8 November 2019 (UTC)

It's an interesting point, though I think you've overstated it a bit. Not all meaningful uses are shorthand. An obvious class of exceptions is the ones that refer to the infinity of the extended real numbers. This is important, for example, in measure theory, where it is completely standard to say that the measure of some set is infinity, and this is not "shorthand" for anything.

That said, as I said on WT:WPM, I do think we need to put some serious thought into exactly what this article is about. I would also support a merger with infinity (philosophy), even though that arguably makes it even harder to figure out what it's about.

I suppose one possibility is to make it a WP:DABCONCEPT. I say that reluctantly because I'm not particularly fond of that class of article. But I'm struggling to figure out what else to do. --Trovatore (talk) 05:46, 8 November 2019 (UTC)

Do not forget that in some (not unusual) contexts of real analysis and topology "infinity" means, literally, the well-known point in the well-known one-point compactification of the real line. Or one of the two points ${\displaystyle \pm \infty }$ in the two-point compactification. And in complex analysis, one-point compactification of the complex plane. In such context, "tends to infinity" becomes, literally, convergence to this point. And "measure equals infinity" is also interpreted literally. As well as "the interval ${\displaystyle [a,+\infty ]}$ etc. (In projective geometry, embedding the plane into the projective plane, we observe rather "the line at infinity".) Boris Tsirelson (talk) 06:42, 8 November 2019 (UTC)

It's also certainly a notable concept, if not a "thing" (which seems to mean an object) that is used in projective geometry and measure theory. — MarkH21 (talk) 07:01, 8 November 2019 (UTC)

Britannica's lead may be useful to consider:

Infinity, the concept of something that is unlimited, endless, without bound. The common symbol for infinity, ∞, was invented by the English mathematician John Wallis in 1657. Three main types of infinity may be distinguished: the mathematical, the physical, and the metaphysical. Mathematical infinities occur, for instance, as the number of points on a continuous line or as the size of the endless sequence of counting numbers: 1, 2, 3,…. Spatial and temporal concepts of infinity occur in physics when one asks if there are infinitely many stars or if the universe will last forever. In a metaphysical discussion of God or the Absolute, there are questions of whether an ultimate entity must be infinite and whether lesser things could be infinite as well.^[1]

Paul August ☎ 11:56, 8 November 2019 (UTC)

All good ideas. Trovatore is surely correct about extended real numbers as are Tsirel and MarkH21 about compactification, projective geometry, and measure theory. If Infinity is redirected to Infinitude the latter will need some {{redirect}} entries such as

"Infinity" redirects here. For Points at infinity, see Extended real numbers and Compactification (mathematics).

The Britannica lead could be adapted to Infinitude but it needs some work. As it stands, it characterizes Infinity as a concept but, in some contexts, Infinity is a measure.

Peter Brown (talk) 19:41, 9 November 2019 (UTC)

I strongly oppose to move the page to Infinitude. The main reason is that this is an article on mathematics and infinitude is rarely used in mathematics. The term is not used in the article. The redirects InfinitudeInfinitude and FinitudeFinitude have each only two links from the main space, and only one of these four links is in an article of mathematics. Wikipedia must reflect the common use; therefore, infinity must be called infinity, even if some people think that it is a misnomer.

About Peter's opening post. This is true that there is no mathematical object called infinity. This does not means that there is no mathematical concept called infinity. On the contrary, the term infinity appears in many mathematical phrases, such "tends to infinity", "point at infinity", "two parallel lines intersect at infinity", "hyperplane at infinity", "an asymptote is a tangent at infinity", "Taylor expansion around infinity", "property at infinity", ... Each of these phrases can be accurately defined without using infinity, but this does not mean that infinity is not a concept. This means that mathematicians are not philosophers, and try to avoid circular definitions. This means also that the concept of infinity has multiple aspects, that cannot be reduced to a single formal definition. Also, many of these phrases were in use a long time before a formal definition could be given.

About Encyclopedia Brittanica quotation. Infinity as a physical concept seems to be an error. Every occurence of infinity that I know in physics can be restated as the question whether the mathematical concept of infinity models correcly the physical reality.

About Peter's opening post. This is true that there is no mathematical object called infinity. This does not means that there is no mathematical concept called infinity. On the contrary, the term infinity appears in many mathematical phrases, such "tends to infinity", "point at infinity", "two parallel lines intersect at infinity", "hyperplane at infinity", "an asymptote is a tangent at infinity", "Taylor expansion around infinity", "property at infinity", ... Each of these phrases can be accurately defined without using infinity, but this does not mean that infinity is not a concept. This means that mathematicians are not philosophers, and try to avoid circular definitions. This means also that the concept of infinity has multiple aspects, that cannot be reduced to a single formal definition. Also, many of these phrases were in use a long time before a formal definition could be given.

About ∞ symbol: It is wrong that it is the mathematical symbol of infinity. It is the symbol of the infinity on the real number line. It is not used for infinite numbers nor for infinity in geometry (points at infinity). Therefore it must be removed from the first sentence of the article, and the section "Infinity symbol" must be adapted. D.Lazard (talk) 12:35, 10 November 2019 (UTC)

I'm convinced. The article must be called "Infinity" unless there's agreement that each the various notions (geometric, cardinal, ordinal, metaphysical, etc.) must be given its own article or article section, in which case Infinity should be a disambuation page.

I do disagree about ${\displaystyle \infty }$. I have never worked with the extended real numbers, but the symbol is very familiar to me as a pseudo-bound in expressions of limit, integration, infinite sums and products, etc. I would naturally read them out loud as "the integral from one to infinity of f of x with respect to x", "the limit of f of x as x tends to infinty", and so forth. I think that it is common to read ${\displaystyle \infty }$ as "infinity". The meaning of an expression is determined by the way it's commonly used, so ∞ is the symbol for infinity. Peter Brown (talk) 14:36, 10 November 2019 (UTC)

I think that D.Lazard point is that while ∞ is a symbol for infinity it is not the symbol for infinity, since in some contexts it is not used. Paul August ☎ 15:15, 10 November 2019 (UTC)

(Outdenting since I have no idea whom I'm replying to at this point). I don't want to get too caught up in all this, but maybe another thought is to have an "Infinity" article which would be a fairly general overview of the very broad concept, including most of what's currently at Infinity (philosophy), along with the stuff from this article from the "Physics" section onward. That article would then include a section on the use of infinity in mathematics, which would be a short summary of this article, which gets moved to "Infinity in mathematics" (along with a {{main}} pointer). The scope of this article can then be pinned down a bit better (infinite sets; infinity as a bound for limits, sums, integrals, etc; point at infinity in compactification; etc etc etc). Does this make sense? Is it a horrible idea? –Deacon Vorbis (carbon • videos) 16:07, 10 November 2019 (UTC)

I think that's actually a pretty good plan. -Trovatore (talk) 08:32, 15 November 2019 (UTC)

In his recent edit, DavidCary is surely correct that the circle, or the ring, is a symbol of infinity (or perhaps eternity). This association definitely merits inclusion in Wikipedia. The choice of the Infinity article is plausible, though other choices might be considered. As Deacon Vorbis says, his contribution is somewhat off-topic for this article, but it is close enough that inclusion merits discussion, at least. It surely doesn't belong in § Arts, games, and cognitive sciences, but perhaps it can be given its own section?

Peter Brown (talk) 21:27, 7 September 2020 (UTC)

(Personally, I think that a circle has nothing to do with infinity. If it does, a square would qualify just as well.)

But seriously, given where the (way too many) sources come from, I don't think it deserves its own section in this article. If it does belong somewhere at all, then indeed the § Arts, games, and cognitive sciences would be the place, provided the section title is amended to § Arts, religion, games, and cognitive sciences - DVdm (talk) 21:45, 7 September 2020 (UTC)

(edit conflict) For better or worse, this article article focuses more on the somewhat more well-defined conceptualization of infinity in mathematics. For the more wishy-washy stuff, we have Infinity (philosophy), but I'd be hesitant to include this even there. The sources I saw gave only brief, off-hand mentions. In order to add something of value, it would have to say a bit about how a circle symbolizes infinity, which would have cultural manifestations other than just wedding rings. –Deacon Vorbis (carbon • videos) 22:04, 7 September 2020 (UTC)

The Dharmachakra and the Ouroboros, respectively from Indian and Egyptian traditions, are circular symbols representing unending (i.e. temporally infinite) cycles. § Arts, games, and cognitive sciences is already too miscellaneous. Philosophy is "wishy-washy"? The field gave rise to logic and natural science.
Peter Brown (talk) 00:41, 8 September 2020 (UTC)

1) Alchemy gave rise to chemistry, but that doesn't stop alchemy from being wonky. But more importantly, 2) Umm, I never even said philosophy was wishy-washy, but rather the article at Infinity (philosophy) deals with the vaguer, more wishy-washy aspects of infinity, whereas this article tends to deal with the more precise mathematical notions. I realize that's maybe not ideal, but it's just the way things are at the moment. –Deacon Vorbis (carbon • videos) 03:10, 8 September 2020 (UTC)

I would dispute that it's "the way things are". Of those mentioned in Infinity (philosophy), whom do you view as wishy-washy? Aristotle? Locke? Certainly, there have been careless philosophers, but they don't make it into Wikipedia. The article has no mention of the alchemists, for example. For thinkers who were unclear on their concepts, the views of l'Hôpital and Bernoulli, mentioned in the more mathematical article, are better examples.

This is of course beside the point. I am arguing in support of DavidCary's thesis that the circle, and I'll add the wheel, are symbols of eternity and infinity. I have mentioned the Dharmachakra and the Ouroboros in support.

@DavidCary:You really need to provide, not just citations, but arguments to show that your citations buttress your thesis. The Doniger book seems totally irrelevant and therefore dilutes the attempt at support. You have been most industrious in coming up with references, but they are mostly from modern authors who assert, without support or much elaboration, that the circle or ring has long been used to represent infinity or eternity. References to Christian literature are particularly doubtful since Christianity's sacred book starts out "In the beginning ...", implying that, unlike a circle, the world had a beginning. Deacon Vorbis refers you to WP:OVERCITE, which I also recommend that you read. Your thesis — which I do agree with — is not well supported by a multiplicity of sources, certainly not any that asserts our view in passing.

Peter Brown (talk) 23:01, 8 September 2020 (UTC)

I am not sure that we really need the infinity symbol in the first sentence, but if it is to remain, I sort of think PolarisBSH makes a decent point. One advantage for having the symbol there is that it's a convenient spot from which to copy the unicode. If we use <math>, that doesn't work. --Trovatore (talk) 23:36, 27 January 2020 (UTC)

Agreed. Unicode is preferable there. - CRGreathouse (t | c) 07:19, 29 January 2020 (UTC)

An editor thought I "preferred" the Latex version... actually, I hadn't seen the Talk discussion about cut-and-paste of the symbol, so pretty much arbitrarily chose the Latex version. In light of this discussion, I've changed it to the Unicode symbol. I see no reason to have three renderings in the lead, nor to point out that one of them is "(in unicode)". --Macrakis (talk) 14:36, 8 September 2020 (UTC)

I certainly killed the "in Unicode" bit (since the {{math}} version was still giving the straight character, just with a different font), but as I said in my edit summary, it's reasonable to present a reader with the various ways they might see it rendered on this page (or even elsewhere on WP). We've got a horrible patchwork of methods for rendering math, and this is just a mild concession to that fact. I've reworded it slightly so it doesn't clutter the opening sentence and to make it clear that it's the same symbol, just rendered differently. –Deacon Vorbis (carbon • videos) 14:49, 8 September 2020 (UTC)

The fact that we have three ways of writing the infinity symbol internally is really not the reader's concern. The extremely minor differences in appearance between the three methods aren't worth mentioning, any more than the article on the letter 'b' says that it may look like b, b, or b in the lead.

I do understand the motivation for having a copyable letter rather than an image, but the reader has no way of knowing that ${\displaystyle \infty }$ is an image rather than a character, so that requirement isn't met.

There is more of an argument for presenting the different glyphs and different ways of writing the symbol in the Infinity symbol article, but even there, using the three internal ways in the lead is unnecessary. --Macrakis (talk) 15:14, 8 September 2020 (UTC)

I also agree that we don't need to show the various renderings and fonts for the infinity symbol in this article. Just pick one copyable symbol. Having three of the same symbol is unnecessary clutter. Readers who are interested in the forms of the symbol itself can go to the article on the symbol. — MarkH₂₁^talk 15:26, 8 September 2020 (UTC)

@Peter M. Brown and Deacon Vorbis: Any objections? --Macrakis (talk) 16:31, 9 September 2020 (UTC)

Yes, but it would appear I'm in the minority here, so I'll probaby just have to live with it. But if you're going to simply use one, it should probably be with {{math}}, as ∞, since it tends to look better. –Deacon Vorbis (carbon • videos) 16:41, 9 September 2020 (UTC)

Not clear that you're in the minority. MarkH21 and I are for choosing 1 symbol; you are for 3; and Brown seems to be for 3 (based on Edit summary). @Trovatore and CRGreathouse: you may wish to comment.

I think we need to be more explicit about our reasoning so that we can discuss productively. Here is mine:

1) Users don't see the varying internal mechanisms, only the visible result.

2) The visible results are very similar, at least as similar as various glyphs for the same alphabetic letter in different fonts. Except in cases where there are structural differences in shape (e.g., the two-counter a vs. the one-counter ɑ), WP doesn't usually comment on such variants.

3) In any case, the shapes will vary based on the OS, browser, and Wikipedia defaults, so it's not as though we can demonstrate the different shapes in a useful way. In fact, in some configurations, they may be identical, which will be even more confusing. If we do need to demonstrate graphic variants (which I don't think we do, certainly not in this article), that can only be done reliably with images, not with textual markup, as is done in the Infinity symbol article, like this:

4) If we do comment about such variants, it's in the article about the symbol (in this case Infinity symbol), not about the concept.

5) We also comment about variants if there is a semantic difference, as in ϕ vs. φ, where the variants (perversely) mean different things in math. But here there is no semantic difference among the variants.

6) As User:Trovatore and User:CRGreathouse point out above, it's a nice courtesy to the user to give a copyable version, so that would be ∞ ({{math|∞}}) or ∞ ({{big|∞}}) rather than ${\displaystyle \infty }$ (<math>\infty</math>).

Agreement? Rebuttal? --Macrakis (talk) 18:05, 9 September 2020 (UTC)

I am for at most one symbol, at least in the first sentence. I'd be OK with zero. I would mildly prefer to make it copyable; no point in frustrating people when you don't have to. But I don't care that much. I do think multiple, nearly identical versions of the symbol in the first sentence is a bad idea, and on that point I care a bit more. --Trovatore (talk) 18:55, 9 September 2020 (UTC)

The lead, with three symbols, scans well. I don't understand why this is a bad idea in itself. It's true, though, that — if there are multiple versions — it's unfriendly to omit indicating anywhere that one can be copied. Once upon a time, the article did say that ∞ was Unicode, as Infinity symbol does now, which indicates to the sufficiently knowledgeable reader (but not to others) that this version could be copied.

∞ is too tiny, however, and ∞ is ugly. Except at Infinity § Symbol, the article uses the better-looking <math>\infty</math> symbol everywhere. Do we need to say, explicitly, that ∞ can be copied, at Infinity symbol if not in the Infinity article itself? Just a suggestion: how about leaving the text of the Infinity symbol article alone but noting in the {{infobox}} legend that the symbol can be copied from there?

Peter Brown (talk) 20:37, 9 September 2020 (UTC)

I really don't think the lead "scans well". Three slightly different versions of the same symbol strikes me as remarkably silly. I don't want to get hyperbolic on this issue because the whole thing doesn't matter that much, but three versions is the worst possible option. --Trovatore (talk) 22:16, 9 September 2020 (UTC)

I concur with Trovatore (on most everything, but in particular) that one symbol should suffice. I think being copyable is important. - CRGreathouse (t | c) 01:21, 10 September 2020 (UTC)

OK, lets use {{math|∞}}. The template discussion calls {{math}} "an alternative to using the <math>...</math> tag pair" so I wasn't distinguishing {{math|∞}} from <math>\infty</math> and didn't realize that the former could be copied. I'm mildly unhappy that it differs from nearly every other use in the article, but let's go with it. Deacon Vorbis, do you want to do the honors, since you've been editing it? Peter Brown (talk) 02:24, 10 September 2020 (UTC)

I'm sort of doing ten things at once right now. I'll look at it tomorrow probably when I can properly fret over the wording, but if anyone wants to in the mean time, that's totally fine. –Deacon Vorbis (carbon • videos) 02:29, 10 September 2020 (UTC)

I've the simplest thing. Perhaps others can improve on it. --Macrakis (talk) 17:48, 10 September 2020 (UTC)

@D.Lazard: The following sentence in the "Geometry" section is a bit strange:

Until the end of the 19th century, infinity occurred rarely in geometry.

I think I understand what you mean, but it's a bit misleading or just incorrect as currently worded. It's not like people didn't use analytic geometry following Descartes (essentially with real numbers, but at least over the rationals or ${\displaystyle {\overline {\mathbb {Q} }}}$) or that they didn't view lines as extending infinitely. — MarkH₂₁^talk 12:51, 2 February 2021 (UTC)

I agree that the sentence is a bit curt and deserves better treatment. But I am not aware of any studies of geometric infinity before the 19th century. Aristotle was clear that lines could be extended indefinitely as needed ("potential infinity"), but he insisted that they never reached "actual infinity." In other words, lines must not be considered to actually extend forever, but you could keep drawing them however long you needed. Mathematicians largely followed his advice up until the 19th century. On the other hand, mathematicians did consider the number of points on a line (segment) to be infinite, even if it did not extend to infinity. Galileo famously pondered the "paradox" of having the same infinite number of points on two segments of different lengths. But he did not resolve the paradox, nor did he claim (as far as I know) that lines extended to infinity. --seberle (talk) 11:20, 8 February 2021 (UTC)

I agree. About the points on a line: As far as I know, the geometers did not considered that a line is composed of points. So, it was not the number of points on a segment that was infinite, but the number of possible positions on the line of a single point. A witness of this is the expression "locus of a point that" instead of "locus of the points that" (the plural was nonsensical before 20th century). D.Lazard (talk) 12:16, 8 February 2021 (UTC)

Now that is interesting! I learned something today; thanks very much. I read my dad's old college math texts when I was a kid, and they used this "locus of a point" phraseology. I never realized it had this significance till today.

Still, infinity in geometry isn't just infinite sets; there's also the point at infinity, and the commonplace that "parallel lines meet at infinity". Our article on projective geometry dates its formalization to the 19th century as well, but what about its pre-formal versions? I'm not great at history and don't have a good sense of how, say, da Vinci might have seen this, in his work on perspective. --Trovatore (talk) 18:28, 9 February 2021 (UTC)

That is true. The only example I know of someone seeming to consider the points (or positions) on a line was in the case of Galileo, which I mentioned above. If I remember correctly, he noticed you could match the points in short segment with the points in a longer segment by drawing segments between them. I think I need to go back to the original and refresh my memory on exactly what he did and whether he referred to "points" on the line or "positions" or what. --seberle (talk) 16:35, 9 February 2021 (UTC)

I have boldy expanded the first sentence into a paragraph. Be free to improve it and/or tagging is with {{citation needed}}. D.Lazard (talk) 18:28, 9 February 2021 (UTC)

The edit looks good. (I'll probably make some minor grammar/style edits.) I still wonder if we should include something on Galileo's Paradox as an exception, showing that some were aware of a primitive idea of an infinite set of points. (Wikipedia has a whole article on Galileo's Paradox.) Galileo did, in fact, repeatedly refer to lines "containing an infinite number of points." --seberle (talk) 07:44, 10 February 2021 (UTC)

Oh, and this entire section is lacking citations. We need to add references. --seberle (talk) 08:36, 10 February 2021 (UTC)

@Peter M. Brown:, the reason why I removed those words is because it says "on the right" which is not the case when viewing the page on mobile device. On phones, the picture is shown just below the text, not on the right side as in a computer. I thought it would create a confusion. ☎️ Churot ^DancePop 16:20, 23 February 2021 (UTC)

Well noted. "... on the right" won't do. Peter Brown (talk) 17:11, 23 February 2021 (UTC)

A discussion is taking place to address the redirect Unboundedness. The discussion will occur at Wikipedia:Redirects for discussion/Log/2021 February 25#Unboundedness until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 21:39, 25 February 2021 (UTC)