# Talk:Infinity symbol

Represents time no longer flowing/running out.  — Preceding unsigned comment added by 47.55.176.9 (talk) 20:08, 9 January 2015 (UTC) 

You have a reliable source for this interpretation? —David Eppstein (talk) 21:18, 9 January 2015 (UTC)
symbolsage.com/eternity-symbols-and-meaning/  octavia
Does not mention hourglasses. And what evidence is there that this web site meets Wikipedia's standards for reliability? —David Eppstein (talk) 05:44, 26 September 2021 (UTC)
In familiy trees the 90 degree rotated 8 is used to indicate 'marriage'. Sorry, maybe I am just the kind of person that needs to be told that the letter 'a' means 'the letter 'a' '.  — Preceding unsigned comment added by 2A0A:A549:7055:0:ED7F:25C5:A476:330D (talk) 05:41, 13 April 2023 (UTC)

Perhaps whoever invented it was considerate for typesetters and chose a form that could be typeset easily as an 8 on its side ("lazy 8"). Anthony Appleyard (talk) 07:38, 22 November 2015 (UTC)
Ok, but why 8 rather than any other symbol that looks different tipped sideways? And have you looked at the original printing of Wallis, as reproduced online? It doesn't seem that he was trying to be kind to typesetters in other ways. Anyway, we should be using reliable sources here rather than speculating ourselves. —David Eppstein (talk) 07:51, 22 November 2015 (UTC)

The Infiniti logo is not a deformed version of the infinity symbol, it's clearly a road pointing in the distance, such as this image http://i.imgur.com/spCswXb.jpg. Don't see the point of including this obscure Lazy 8 studios, given that there are plenty of other more notable examples without it, and their Wikipedia page doesn't even show the logo. It isn't supposed to be an exhaustive list of companies employing the infinity symbol in their logo.
CrocodilesAreForWimps (talk) 04:22, 15 July 2016 (UTC)

I don't see why the Infiniti logo can't be both a deformed infinity symbol and an image of a road leading to infinity, at the same time, a visual pun. But since this is obviously contentious, we need reliable sources and all I can find are unreliable and contradictory web sites and discussion forums. Unless someone can find sources, we're better off without this; it's not needed to make the point that the symbol is a frequent subject of logo design. —David Eppstein (talk) 06:25, 15 July 2016 (UTC)
Doesn't make sense to look at it as a deformed ∞, when there's a very obvious meaning in it that conveys the concept of infinity in the context of driving. If other people have the opinion it can be interpreted differently that's nice, but it has no place without a reliable source, as you say. CrocodilesAreForWimps (talk) 19:02, 15 July 2016 (UTC)
This source appears to be reliable, and confirms that the symbol has more than one meaning. But the two meanings it gives are the road one and Mt. Fuji, not the infinity symbol. —David Eppstein (talk) 20:19, 15 July 2016 (UTC)

This unreferenced statement is most likely false. Please read the section Typesetting#Pre-digital era and look at "Diagram of a cast metal sort." Also, look at the full-sized photo of "Movable type on a composing stick on a type case" (photo at top of article). In both of these, you will see that a cast metal sort (a letter or character of moveable type) is rectangular in shape. Hence, there is no way that you can put a cast metal sort for an "8" on its side to obtain the infinity symbol. It would take a separate cast metal sort to obtain this symbol. 
In 2011, I removed a similar sentence from Infinity. For the 2011 reply to my above statement, see Talk:Infinity#Removed unreference statement about turning an "8" on its side. --RJGray (talk) 16:13, 6 September 2017 (UTC)

Actually, the cast metal letters and symbols were rectangles or squares of various sizes, the illustration in Typesetting#Pre-digital era is just an example. And the pages of some mathematical journals were composed manually until very recently in some countries (in India for instance; I have under the eyes a reprint from the Indian Journal of mathematics (Allahabad) of 1990, in which it is clearly the case; it is quite a work of art!). So it is perfectly possible that some symbols were sometimes rotated. But I agree that it would need to be sourced. Sapphorain (talk) 17:15, 6 September 2017 (UTC)

The first sentence of the Usage sentence is as follows:
"In mathematics, the infinity symbol is used more often to represent a potential infinity, rather than an actually infinite quantity as included in the extended real numbers, the ordinal numbers and the cardinal numbers (which use other notations). For instance, in mathematical expressions with summations and limits such as the one below:

  
    
      
        
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    {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}}}=\lim _{x\to \infty }{\frac {2^{x}-1}{2^{x-1}}}=2,}
  

the infinity sign is conventionally interpreted as meaning that the variable grows arbitrarily large towards infinity—rather than actually taking an infinite value."
This is complete nonsense.
There is no such things as a "potential infinity" in mathematics.
The infinity symbol as the upper limit of the summation sign signifies an actual infinity: the number of terms in the summation.
Nonexistent concepts such as "potential infinity" do not need to be invented and cited. 2601:200:C000:1A0:A092:B45C:E058:ACB4 (talk) 00:37, 3 November 2021 (UTC)

Potential infinity is a well-studied concept, not one invented recently. However, I'd like instead to pick out a very serious error in your comment. The upper limit of a summation sign does not conventionally denote the number of terms in a summation; when finite, it denotes the largest term. When written as the infinity symbol, it does not denote the fact that there are infinitely many terms, but rather the fact that there is no largest term. The summation is not taken to include an infinite term, in such cases. —David Eppstein (talk) 01:17, 3 November 2021 (UTC)
I would not agree that there is no such thing as potential infinity in mathematics: a standard sequence, i.e. an endless chain, is such a thing.  OTOH, actual infinity has to do with fixpoints or limits (aka "completed infinity"): and, in that sense, that "In mathematics, the infinity symbol is typically used to represent a potential infinity. For instance, in mathematical expressions with summations and *limits* such as [follows an infinite sum which is indeed *the limit of* a series of partial sums]" is indeed utter nonsense... LudovicoVan (talk) 11:02, 3 May 2025 (UTC)
The value of the limit is a kind of completion. But the infinity symbol is not used as that value, but as a boundary towards which the terms of the sum or limit progress without actually reaching. In that sense, it is a potential infinity and the references in the article describe it as that regardless of your personal opinion, which is far less relevant here than what the references say. —David Eppstein (talk) 18:12, 3 May 2025 (UTC)
I see your point (your "opinion" I should say) and I strongly disagree: the value of the limit indeed *is* the limit, there is no approximation or "kind of" going on, which is true regardless of underlying foundations, i.e. even an epsilon-delta is about completion and nothing else.  And that's it, the "infinity" symbol really stands and only stands for some *completed* infinity: in fact, in my opinion, this article rather re-injects and totally for free the confusion of the notions of infinity that is typical and only typical of the mainstream treatment of standard set theory...  What is an arbitrary union over an index set?  I won't be allowed to touch the article of course, but here is a "mathematical tip" for the innocent if you won't mind: informally but not too much, `lim_{n->oo} n = oo`. LudovicoVan (talk) 20:31, 3 May 2025 (UTC)
Indeed, it's the arrow in `n->oo`, as opposed to a plain equality as in `n=oo`, that expresses the idea of approximation/asymptotics, not the infinity symbol 'oo'. LudovicoVan (talk) 10:50, 7 May 2025 (UTC)
There is no arrow in summation notation. —David Eppstein (talk) 01:30, 8 May 2025 (UTC)
Thanks for all the fish. LudovicoVan (talk) 09:54, 8 May 2025 (UTC)

I restored the mention of Euler’s variant of Wallis symbol’s of infinity. I think the mention of it is relevant to this page, and especially so since Euler is very famous. The fact that we don’t have secondary sources confirming this variant didn’t catch on appears to me as a further argument to mention it, as the primary sources (Euler’s papers), are a proof it was indeed used. Wikipedia cannot restrain from using information on the ground that it is not confirmed by a secondary source: it would mean Wikipedia is a second rate encyclopedia.--Sapphorain (talk) 20:50, 4 January 2022 (UTC)

User:Sapphorain has now twice reverted my removal of the material claiming that Euler used a variant of this symbol, adding off-topic material about Euler's use of the mathematical concept of infinity rather than of this symbol specifically, separately claiming that this variation was not used by anyone else, and separately claiming that this variant resembles a totally-unrelated symbol in Unicode for something in Japanese, sourced only to Euler's own works. The material does not contain any secondary sources attesting to the significance of this variation in the history of the symbol. It does not have sources backing up the implication that this was a deliberate and significant change to the symbol rather than just a minor typographic variant, much like different fonts with this symbol today vary in having constant or variable line width, equal or unequal lobe size, and broken or unbroken lines at the crossing point. And it does not have sources proposing the vaguely-similar Japanese symbol as a way to display something resembling Euler's symbol. I am trying to get the article in shape for a Good Article nomination and Sapphorain's insistance on including unsourced original research is a significant obstacle to this effort. Does anyone besides Sapphorain think it should be there? Alternatively, can any of this material be properly sourced? —David Eppstein (talk) 20:56, 4 January 2022 (UTC)
I just wish to mention here, for the sake of precision, that User:David Eppstein has now twice reverted my original inclusion of the mention of Euler's variant. I will of course respect a consensus on that matter.--Sapphorain (talk) 21:11, 4 January 2022 (UTC)
Since you mention precision, you are being incorrect and imprecise. I did not twice revert your edit. I removed your material once, not as a specific revert of its addition, but as part of a much larger effort to revise the article and clean out cruft from it. In doing so I did not look at the edit history and did not pay any attention to when it was added, who added it, or whether it was added in a single edit. I then reverted your re-addition of the material, once. —David Eppstein (talk) 21:17, 4 January 2022 (UTC)
I agree with David's removal of the text written by Sapphorain: it is not only poorly sourced, but also contains considerations on Euler's views of infinity, that do not belong to this article, even if they were correctly sourced. However, the figure could be restored with a caption such as "typographic variant of the symbol that was used by Euler". D.Lazard (talk) 21:23, 4 January 2022 (UTC)
Well, maybe, if it were a totally different image that actually looked like Euler's symbol, if it fit among all the other images in the article, and if we had sources documenting whether it was intended as a variant, the same symbol just looking different because of how it was typeset, or a different symbol altogether. I'd rather find a way of squeezing in the Métis flag. —David Eppstein (talk) 22:41, 4 January 2022 (UTC)
I had never heard of this variation of the symbol before, but Cajori, A History Of Mathematical Notations Volume II §421 [1] does talk about it, in one sentence. So I guess it's reasonable to mention it in an article about the symbol, without over-emphasizing it. (By the way, there's some other interesting stuff in that section too, like the notation 
  
    
      
        
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    {\displaystyle \infty ^{n}}
  
, which I see was discussed at MathOverflow before.) Adumbrativus (talk) 02:29, 5 January 2022 (UTC)

I agree completely with David Eppstein's criticism of Sapphorain's edit. But based on this Cajori source, I also agree with Adumbrativus that it would be perfectly reasonable to replace Sapphorain's three sentences by a single one along the lines of something like "Various mathematicians, including Leonhard Euler, have used certain typographical variants of the symbol, particularly an open version." But it would also be fine to remove altogether. Gumshoe2 (talk) 03:01, 5 January 2022 (UTC)
So first let me say thank you to Sapphorain and Adumbrativus for the delightful links!  I guess the Cajori one is in the public domain?  And the original Euler piece had some very enjoyable stuff:  Expressiones ergo 
  
    
      
        
          
            
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    {\displaystyle {\frac {2\cdot 3\cdot 7\cdot 11\cdot 13\cdot {\mathit {etc}}}{1\cdot 2\cdot 6\cdot 10\cdot 12\cdot {\mathit {etc}}}}}
  
 valor est infinitus, et posito absolute infinito 
  
    
      
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    {\displaystyle =\infty }
  
, erit istius expressionis valor 
  
    
      
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    {\displaystyle =l\infty }
  
, quod infinitum inter omnes infiniti potestates est minimum. (Here I've used the ordinary ∞ because I don't have the variant handy.) I guess the 
  
    
      
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    {\displaystyle l\,\!}
  
 means logarithm, and the point is that the infinite product diverges very slowly?  But why is it the smallest?  Wouldn't the double logarithm be smaller, looking at things this way? 
Anyway, neither here nor there.  David is quite right that we shouldn't be inferring something like this from a possibly oddly typeset primary source.  But now that there's a secondary source, I think it would be nice to mention it.  Not everything we do has to be grim and serious. --Trovatore (talk) 07:06, 5 January 2022 (UTC)
My guess at the terminology is that "=l" means "equals in the limit". He seems more careful to distinguish this from the usual kind of equality than I might have supposed, given the stereotype of his sloppy manipulations of series. —David Eppstein (talk) 08:35, 5 January 2022 (UTC)
Hmm, but then why would he say that it's the smallest infinite among all infinite powers? --Trovatore (talk) 16:24, 5 January 2022 (UTC)
Euler's paper has an English version.--SilverMatsu (talk) 15:04, 5 January 2022 (UTC)
Maybe, but you don't seem to have linked to it :-).  In any case the Latin seems quite readable. --Trovatore (talk) 16:25, 5 January 2022 (UTC)
Sorry, the correct link is this(PDF). The link I wrote earlier was the original Latin version.--SilverMatsu (talk) 00:39, 6 January 2022 (UTC)

@David Eppstein, I have a question regarding these series and limit expression in this section. I couldn't find any relation between this expression

  
    
      
        
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    {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}}}=\lim _{x\to \infty }{\frac {2^{x}-1}{2^{x-1}}}=2}
  

Also, I haven't found yet that expression in a cited reference, especially for that limit. Would you like to give an explanation? Regards, Dedhert.Jr (talk) 10:26, 17 August 2022 (UTC)

This article is not about the truth of these equalities, but about their use of 
  
    
      
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    {\displaystyle \infty .}
  
 Nevertheless, the fact that the sum of the series is 2 is proved in geometric series and 1/2 + 1/4 + 1/8 + 1/16 + ⋯. The value of the limit results from the equality 
  
    
      
        
          
            
              
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    {\textstyle {\frac {2^{x}-1}{2^{x-1}}}=2-{\frac {1}{2^{x-1}}}}
  
 and the basic properties of exponential function. D.Lazard (talk) 13:26, 17 August 2022 (UTC)
Yes, pretty much. The limit is of partial sums of the series. They were chosen merely to display the notation rather than to have any deep mathematical significance. I think WP:CALC allows us to include such material without requiring a source. —David Eppstein (talk) 16:05, 17 August 2022 (UTC)

Tap the ?123 button and tap =\< button and go to the = and hold it for 3 sec and go to 1 left and you can type infinity method: 2 type ∞ in the google search and go find the symbol in the text blue it and tap copy and go to your keyboard and tap the text thing for 0.0167-1 sec and tap paste and ∞ method 3: go to settings and go to symbols and insert =(but holded and holded 1 left) and type in the name the name of the symbold and go to keyboard and type somthing you named it and it will apper and this won't update i will just let you guys enjoy and those are just the 3 methods to type it 2601:680:8301:73E0:94BC:EB38:87BF:6E05 (talk) 06:14, 2 January 2024 (UTC)

Or you could get a Mac and just hold down the option key while typing 5. ∞. —David Eppstein (talk) 06:47, 2 January 2024 (UTC)
I have no idea how to type that symbol, so maybe you could just literally copy-and-paste it? But meh! Whatever! Dedhert.Jr (talk) 07:38, 2 January 2024 (UTC)

"In mathematics, it often refers to infinite processes (potential infinity) rather than infinite values (actual infinity)."
In this sentence, potential infinity links to the same article as actual infinity. This is incredibly confusing. 
Aaron Bruce (talk) 01:09, 22 April 2025 (UTC)

It's because that article is where the distinction between actual infinity and potential infinity is discussed. —David Eppstein (talk) 01:27, 22 April 2025 (UTC)
Would it make sense on the linked page that the bit about potential vs actual infinity be it's own section and potential infinity on this page could link directly to it? Aaron Bruce (talk) 01:55, 22 April 2025 (UTC)
Isn't that the entire article? —David Eppstein (talk) 02:12, 22 April 2025 (UTC)

Infinity is not a natural or real number, but a concept and notation. This is a well-known fact to mathematicians and common misconception to the general public. It is especially confusing, because infinity can be defined in certain extensions of the real numbers. As David Eppstein correctly points out, infinity is not a "church key, a Wikipedia editor, nor a species of seabird". But these properties are not relevant to the article. If used as a natural or real number in algebra, contradictions can be derived from infinity similar to division by zero.
I am frustrated by snipe reverts without constructive reason. Please revert the revert. Benedikt Schöps (talk) 20:56, 22 July 2025 (UTC)

No, in an infinitary system, infinity *is* a number, in the sense of a mathematical object proper.  Indeed, that includes any system where infinity is a "constant": e.g. the infinities of standard floating-point *are* numbers (in fact, in floating-point, even NaN strictly speaking is a number). LudovicoVan (talk) 22:01, 22 July 2025 (UTC)
The issue I have with your addition is that the sentence seemed like a non sequitur and broke up the narrative flow of the paragraph. This article already discusses the various interpretations of this symbol in § Mathematics, and the one sentence summary in the lead describes that just fine in my opinion; it could plausibly be expanded, but you should work on how to write that and perhaps make a proposal here before making another change directly to the text. In my opinion further detail about whether infinity is one or another type of number belongs at the linked articles Infinity and Actual and potential infinity, where it can be unpacked in a meaningful way.
(Aside: I don't find "it is also not a church key, a Wikipedia editor, nor a species of seabird" to be a particularly helpful revert summary. If we reject contributions it should be clear and forthright without jokes or sarcasm.) –jacobolus (t) 22:51, 22 July 2025 (UTC)
I agree that the edit summary was not very informative. The real reason was that interjecting "it's not a number" at that point in the article confuses rather than clarifies the point that was being made there about the distinction between actual and potential infinity. Additionally the phrasing used for the addition implied that infinity is some specific and well-defined mathematical object that can be said to be not a number; but instead, this article is about a symbol with multiple uses and meanings some of which may (actual infinity) or may not (potential infinity) represent a specific mathematical object. —David Eppstein (talk) 23:02, 22 July 2025 (UTC)
Having given a somewhat similar edit summary at infinity, I'd like to say a couple words here.
Benedikt Schöps, you need to be aware that this is not the first time this issue has come up.  There seems to be a contingent that is very desirous of stating, somewhere in these articles, something their secondary-school math teachers likely said, namely "infinity is not a number".  This is a proposition whose truth value cannot be evaluated, because it contains two under-specified terms, "infinity" and "number".  Exactly why we should be so careful to state this is usually not made clear.  I suppose it's to head off cases of naive equation-manipulation where someone claims, for example, to find a solution by substituting infinity (which is sometimes a correct action in an appropriate context and sometimes not).  But it is not the purpose of Wikipedia articles to anticipate and prevent all possible mistakes.
You on the other hand at least specified "number" well enough that the assertion is clearly true, but in doing so you've robbed it of most of its (already tenuous) point.  A reader who is sophisticated enough to be thinking of "natural numbers" or "real numbers" already knows that those do not include infinity. --Trovatore (talk) 18:47, 23 July 2025 (UTC)
There was actually a (related) point to my edit summary, by the way: it's rarely helpful to define something by what it is not. There are too many things that something might not be, that you never learn what it actually is. —David Eppstein (talk) 19:04, 23 July 2025 (UTC)
Yes, agreed. --Trovatore (talk) 19:32, 23 July 2025 (UTC)


### Potential infinity in math section
Having seen Jacobolus's reference above to the Mathematics section of this article, I went and looked at it.  I think this may need some adjustment.
The text currently asserts [i]n mathematics, the infinity symbol is typically used to represent a potential infinity.  For some value of "typically", that is probably true.  But I'm concerned that it tends to downplay the important cases in which the symbol refers to an actually infinite quantity.  The most obvious case is measure theory, where for example the Lebesgue measure of a subset of the plane may be +∞, indicating that the area of the subset is actually infinite. --Trovatore (talk) 20:00, 23 July 2025 (UTC)

Could say "often" rather than "typically". But I'm not entirely sure that your example does have an actual infinite value. The Lebesgue measure is defined as an infimum and saying that it is +∞ can be thought of as meaning that this infimum has no finite bound (potential infinity) rather than as meaning that it has a well-defined specific value that is an infinite number (actual infinity). And in this case it's not even an infimum of a well-defined set of reals, but rather of a family of sums that do not converge, so it's doubly not defined rather than having a specific value. —David Eppstein (talk) 20:04, 23 July 2025 (UTC)
No, I don't agree.  Measures are typically defined with codomain [0, ∞].  See Folland. --Trovatore (talk) 20:25, 23 July 2025 (UTC)
Ok, I added measure theory as another context where this is used for an actual infinity, and weakened the statement about it "typically" being a potential infinity, per your comments with this reference. Please check my new additions. —David Eppstein (talk) 21:27, 23 July 2025 (UTC)
Looks good.  I changed "unmeasurable" to "nonmeasurable", which I think is more standard. --Trovatore (talk) 21:54, 23 July 2025 (UTC)
There's also 
  
    
      
        
          
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    {\displaystyle \mathbb {R} ^{\infty }}
  
 to mean the set of all infinite-tuples in 
  
    
      
        
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    {\displaystyle \mathbb {R} }
  
. See Sequence space.  – Farkle Griffen (talk) 00:49, 25 July 2025 (UTC)
Do you think that is an actual or potential infinity? If actual, what mathematical object does the exponent represent? If it is intended to represent a space of functions from an enumerated set to 
  
    
      
        
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    {\displaystyle \mathbb {R} }
  
, with the 
  
    
      
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    {\displaystyle \infty }
  
 standing in for the enumerated set, I would have thought the notation would be something like 
  
    
      
        
          
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    {\displaystyle \mathbb {R} ^{\mathbb {N} }}
  
 or maybe 
  
    
      
        
          
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    {\displaystyle \mathbb {R} ^{\omega }}
  
. —David Eppstein (talk) 01:10, 25 July 2025 (UTC)
Actual. It comes from the fact that 
  
    
      
        
          
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 means "n-dimensional space", or the set of all n-tuples in R. So 
  
    
      
        
          
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 means literally infinitely many dimensions, or the set of infinitely long tuples (sequences). You're right that 
  
    
      
        
          
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    {\displaystyle \mathbb {R} ^{\mathbb {N} }}
  
 is also common, and I think Munkres uses 
  
    
      
        
          
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    {\displaystyle \mathbb {R} ^{\omega }}
  
, but 
  
    
      
        
          
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    {\displaystyle \mathbb {R} ^{\infty }}
  
 is still pretty common. – Farkle Griffen (talk) 01:21, 25 July 2025 (UTC)
After thinking about this overnight I think that rather than shoehorning this into the actual/potential infinity distinction we might want a third paragraph on the use of 
  
    
      
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    {\displaystyle \infty }
  
 as a subscript or superscript, where it does not really denote an object itself but is used to flag some infinite characteristic of the object that the full notation denotes. This could apply also to 
  
    
      
        
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    {\displaystyle L^{\infty }}
  
 as well as 
  
    
      
        
          
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    {\displaystyle \mathbb {R} ^{\infty }}
  
. But it would have to be carefully worded and sourced to avoid original research about the philosophy of this notation. —David Eppstein (talk) 17:02, 25 July 2025 (UTC)
The distinction between actual and potential infinity does not belong to this article, and must not be mentioned in section § Mathematics. Indeed, this is a philosophical distinction, not a mathematical one. For example, in Infinity#Real analysis, there are various occurrences of ⁠
  
    
      
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    {\displaystyle \infty }
  
⁠ that can be interpreted as actual or potential infinity, depending on reader's intepretation of notation. In particular, the same ⁠
  
    
      
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    {\displaystyle \infty }
  
⁠ can be interpreted both as a unbounded limit (potential infinity) or an element of the extended real number line (actual infinity).
So, I strongly suggest to remove all mentions of actual or potential infinity in § Mathematics. I would not oppose to the creation of a section Infinity symbol § Philisophy, although such a section would better be placed in Infinity. D.Lazard (talk) 14:50, 25 July 2025 (UTC)

This ignores centuries of tradition where mathematicians were very adamant about limts/sums/series not being actual infinity. Gauss, for instance, said: "I protest above all against the use of an infinite quantity as a completed one, which in mathematics is never allowed. The Infinite is only a manner of speaking." – Farkle Griffen (talk) 15:16, 25 July 2025 (UTC)
Additionally, as this is an article about notation rather than about the mathematics itself, I think the philosophical issues are relevant and important in helping to understand why certain notations might have been chosen for certain concepts. —David Eppstein (talk) 17:05, 25 July 2025 (UTC)
I sort of see Prof Lazard's point here.  As you say, this is an article about the notation, and for that reason seems (to me at least) further removed from philosophy than the mathematics is.  An article about a symbol is a thin reed on which to balance ontological concerns.
That said, I don't think a brief mention of potential and actual is out of line, and I'm not sure I see much to trim from the existing text.  Just let's not let it expand further. --Trovatore (talk) 00:15, 26 July 2025 (UTC)