# Talk:Cantor's first set theory article

Thanks to Carl and William M. Connolley for their insightful comments on the sentence that has been removed. I wrote that sentence and now regard it as poorly written. So I'm pleased that it was removed. However, a sentence is needed here because a lead should mention all topics covered in an article.
With the aid of Carl and William's comments, I wrote a new sentence that eliminates two flaws. First, the new sentence just mentions the analysis of Cantor's article and not its conclusion. It's not the job of the lead to announce the conclusion of a detailed analysis. Later, the article discusses this conclusion and the disagreement about Cantor's proof. Second, I've added a reference to an article containing an analysis similar to the one presented in this Wikipedia article. --RJGray (talk) 21:39, 12 April 2016 (UTC)

I've been looking over the references in the lead using WP:LEAD#Citations, which says (among other things): 

"The verifiability policy advises that material that is challenged or likely to be challenged, and direct quotations, should be supported by an inline citation."
"Because the lead will usually repeat information that is in the body, editors should balance the desire to avoid redundant citations in the lead with the desire to aid readers in locating sources for challengeable material."
"The necessity for citations in a lead should be determined on a case-by-case basis by editorial consensus."
I removed one ref (1st sentence, 2nd paragraph) because the presence of Cantor's existence proof in the article is not controversial, it's well covered by the refs in the following two sentences (which include text about the proof), and it's redundant (it shows up twice in "The article" section).
The ref at the end of the first paragraph is also a candidate for removal since it doesn't seem controversial. There's one ref in the 3rd paragraph. I'm leaning towards keeping it since it's a bit surprising, not commonly known, and did not appear in the literature until 1976. Anyone have thoughts about these two refs or about removing other refs or adding refs in the lead? Thanks, --RJGray (talk) 19:28, 29 April 2016 (UTC)

My general feeling is that we need references in the lead either for direct quotes or when the lead says something that is not repeated in more depth later. So for instance the quote "Cantor's revolutionary discovery" in the lead needs a reference, but the last sentence of the first paragraph (the claim about what the title refers to) doesn't, because it is expanded in more detail in the "influence of Weierstrass and Kronecker" section. —David Eppstein (talk) 20:15, 29 April 2016 (UTC)
Thank you David for your guidance on refs in the lead. I've removed the ref in the last sentence of the first paragraph. I've also re-evaluated the lone reference in the 3rd paragraph. My reasons for keeping it don't fit into your general feeling about refs in the lead since what it refers to is also covered in detail in the "influence of Weierstrass and Kronecker" section. My reasons for keeping it are also not mentioned in WP:LEAD#Citations. So I'm planning to remove this ref in a few days unless someone comes up with a good reason for it. After it's removed, all the refs in the lead will be unique to the lead. --RJGray (talk) 15:19, 4 May 2016 (UTC)

In the following sentence, by shrinking your window, in the first use of nowrap, you can get the subscript 1 in "an1" or one of the other subscripts to wrap to the next line. In the second use of nowrap, you can get the comma in "nν," to wrap to the next line. Is there anywhere we can submit a bug report on nowrap? I switched over to "&nbsp;". 

In his letter introducing the concept of countability, Cantor stated without proof that the set of positive rational numbers is countable, as are sets of the form (an1, n2, …, nν) where n1, n2, …, nν, and ν are positive integers.
In the following sentence, nowrap is necessary because sfrac can cause a wrap after a "(" or before a ")":

The function can be quite general—for example, an1, n2, n3, n4, n5 = (⁠n1/n2⁠)⁠1/n3⁠ + tan(⁠n4/n5⁠).
I will be checking all uses of nowrap and sfrac in this article. --RJGray (talk) 16:20, 4 May 2016 (UTC)

Yes, nowrap and similar not-supposed-to-wrap markup including {{math}} have been broken (at least on Chrome) for several weeks now. I'm not sure whether it's a Wikipedia bug or a browser bug. —David Eppstein (talk) 17:49, 4 May 2016 (UTC)
PS I see you've tried replacing them with &nbsp;. I think that is similarly broken. —David Eppstein (talk) 18:50, 4 May 2016 (UTC)
Actually, I've got them working with &nbsp; except in the one case of a subscript 2n – 1 where I had to use a nowrap within the  <sub> …</sub> . So I've managed to work around the problem. I've went into Internet Explorer and the nowrap in my example above does work there so it might just be a Chrome bug. Also, the sfrac problem is in both Chrome and Internet Explorer so that problem seems to be a Wikipedia bug. RJGray (talk) 20:25, 4 May 2016 (UTC)

Thank you David for telling me that I may be seeing a bug. I've done some more experimenting and found a sentence that I can get to wrap properly in either Chrome and Internet Explorer, but not both.
The following sentence which uses "&nbsp;" wraps properly in Chrome, but not in Internet Explorer. For example, by shrinking your window, the 3 occurrences of "g(" can wrap before the "(".
A one-to-one correspondence between T and R is given by the function: g(t) = t if t ∈ T0, g(t2n – 1) = tn, and g(t2n) = an.
The next sentence which  uses "nowrap" wraps properly in Internet Explorer, but not in Chrome. The 1st occurrence works for Chrome, but Chrome can wrap within the subscripts in the next 2 occurrences. 
A one-to-one correspondence between T and R is given by the function: g(t) = t if t ∈ T0, g(t2n – 1) = tn, and g(t2n) = an.
I've just fixed the 1st occurrence so that at least it works in both browsers. In many cases "&nbsp;" works in both browsers, which is why I've been switching over to it.
I find these examples very interesting. I am specifying the same no-wrap regions in two different ways. Since I'm getting two different behaviors in Chrome or Internet Explorer, it seems unclear whether the bug is in a browser or in Wikipedia code. However, since I can get the proper no-wrapping behavior by using different text, the problem can be fixed in Wikipedia. Actually, I could do it myself with we had a browser template so I could write:  {{browser | Chrome | … }} {{browser | Internet Explorer | … }} {{browser | default | … }} . Of course, it would be preferable for this to be done by the Wikipedia people who maintain "&nbsp;" and "nowrap". Can my examples be communicated to them?  Thanks, RJGray (talk) 14:31, 6 May 2016 (UTC)

@RJGray: Is this still an issue now? There is a system for reporting bugs, but it's changed since last time I did that, so I'd have to look into how it's done all over again. Michael Hardy (talk) 21:14, 14 September 2018 (UTC)

Can someone explain how the initially-hidden content of this article (e.g. the list of algebraic numbers, proof of the value for the method for generating an irrational, etc) is in compliance with MOS:DONTHIDE? Because it doesn't seem to be to me. (Also the square roots in the list of algebraic numbers are horribly formatted in android mobile view.) —David Eppstein (talk) 05:44, 2 July 2018 (UTC)

I still think the table of algebraic numbers is merely supplementary, similar to the past-years statistics example given in MOS:DONTHIDE. The enumeration proof for algbraic numbers can be understood without uncollapsing the table, but it gives some impression about how the enumeration looks like. Similarly, as far as I remember, the value ot the constructed irrational number is just supplementary; its irrationality has been proven before. Probably RJGray knows the details. —
Concerning the root formatting: I remember I had used a unicode "√" followed by (e.g.) "{{overbar|5}}" to generate "√5". Subsequently, sombody has changed this to "{{radic|5}}", which generates "√5".Both look the same on my PC; I can't compare them in android mobile view. Anyway, it appears to be a problem of the "{{radic}}" template now. - Jochen Burghardt (talk) 11:19, 2 July 2018 (UTC)

Ipsic asks the following: Explain what seems to me to be a subtle point. Why doesn't Cantor's second theorem apply to a world of purely algebraic numbers?
Here's the text that came with this request: (This text was in the section "Second theorem", just after the paragraph starting with "The proof is complete since ...")
Note, that under cases 1 and 3, above, the real number in [a, b] that is not a contained in the sequence may be chosen to be any of an infinite number of algebraic numbers that are contained within the intervals (aN, bN) or  [a∞, b∞], respectively.  However, in case 2 where  a∞ = b∞, there is no interval from which an arbitrary algebraic number may be chosen.  The value of a∞ must not be algebraic, because asserting that it is algebraic leads to a contradiction with the first theorem.
I agree that this can be a subtle point. The important point is to distinguish what Cantor's theorem does in general from what it does in a particular application, such as being applied to the sequence of all algebraic numbers in an interval. In general, Cantor's theorem only guarantees that the real number obtained from his construction is not in the given sequence. In many applications, such as applying it to the sequence of rationals in (0, 1) as in the section "Example of Cantor's construction", you are not guaranteed to obtain a transcendental number but in this case you are guaranteed to obtain an number that is not rational. Applying the theorem to this case, you obtain 
  
    
      
        
          
            2
          
        
        −
        1
      
    
    {\displaystyle {\sqrt {2}}-1}
  
, which is a non-rational algebraic number.
However, if you apply Cantor's theorem to the sequence of all algebraic numbers in an interval (a, b), you are guaranteed to obtain a non-algebraic number in the interval.
Since you brought up that you can hit an algebraic number in Case 1 or Case 3, I wish to point out that the section "Dense sequences" proves that a dense sequence—such as, the sequence of algebraic numbers in an interval—never ends up in Case 1 or Case 3, so your argument fails here. To obtain a transcendental number, Cantor is using such a dense sequence.
If you still find this confusing, please let me know. By the way, if you have questions about an article, the questions really belong on the Talk pages and not in the article's text. --RJGray (talk) 02:00, 13 September 2018 (UTC)

The article currently says:

Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved.
There is a cite, which quotes Sheppard:

Cantor's proof of the existence of transcendental numbers is not just an existence proof. It can, at least in principle, be used to construct an explicit transcendental number.
and Stewart:

Meanwhile Georg Cantor, in 1874, had produced a revolutionary proof of the existence of transcendental numbers, without actually constructing any.
But these two statements are not actually in conflict, and we have no evidence that Sheppard and Stewart actually "disagree" in the slightest.  Sheppard correctly notes that Cantor's method can be applied to find a particular transcendental; Stewart correctly notes that Cantor did not in fact do that.
I'm not sure how to fix this, but I don't think it can stand as is.  There is no modern "disagreement" on the question, not phrased this way.  (I'm not sure whether intuitionists consider the proof constructive, because it might use excluded middle (?) but that's a bit of a different issue.) --Trovatore (talk) 07:05, 13 September 2018 (UTC)
Thank you for your feedback. I agree with you that "Sheppard correctly notes that Cantor's method can be applied to find a particular transcendental." However, I disagree that "Stewart correctly notes that Cantor did not in fact do that."
To understand what Cantor did, I'll quote from the article Georg Cantor and Transcendental Numbers . From the bottom of page 819 to page 820, it states:

Cantor begins his article by defining the algebraic reals and introducing the notation: (ω) for the collection of all algebraic reals, and (ν) for the collection of all natural numbers. Next he states the property mentioned in the article's title; namely, that the collection (ω) can be placed into a one-to-one correspondence with the collection (ν), or equivalently:
... the collection (ω) can be thought of in the form of an infinite sequence: (2.) ω1, ω2, ...,  ωη, ... which is ordered by a law and in which all individuals of (ω) appear, each of them being located at a fixed place in (2.) that is given by the accompanying index.
Cantor states that this property of the algebraic reals will be proved in Section 1 of his article, and then he outlines the rest of the article:
To give an application of this property of the collection of all real algebraic numbers, I supplement Section I with Section 2, in which I show that when given an arbitrary sequence of real numbers of the form (2.), one can determine [Note: By this Cantor means that he can "construct"] in any given interval (α ··· β), numbers that are not contained in (2.). Combining the contents of both sections thus gives a new proof of the theorem first demonstrated by Liouville: In every given interval (α ··· β), there are infinitely many transcendentals, that is, numbers that are not algebraic reals. Furthermore, the theorem in Section 2 presents itself as the reason why collections of real numbers forming a so-called continuum (such as, all the real numbers which are ≥ 0 and ≤ 1), cannot correspond one-to-one with the collection (v); thus I have found the clear difference between a so-called continuum and a collection like the totality of all real algebraic numbers.
To appreciate the structure of Cantor's article, we number his theorems and corollaries:
Theorem 1. The collection of all algebraic reals can be written as an infinite sequence.
Theorem 2. Given any sequence of real numbers and any interval [α, β], one can determine a number η in [α, β] that does not belong to the sequence. Hence, one can determine infinitely many such numbers η in [α, β]. (We have used the modern notation [α, β] rather than Cantor's notation (α ··· β).)
Corollary 1. In any given interval [α, β], there are infinitely many transcendental reals.
Corollary 2. The real numbers cannot be written as an infinite sequence. That is, they cannot be put into a one-to-one correspondence with the natural numbers.
Observe the flow of reasoning: Cantor's second theorem holds for any sequence of reals. By applying his theorem to the sequence of algebraic reals, Cantor obtains transcendentals. By applying it to any sequence that allegedly enumerates the reals, he obtains a contradiction—so no such enumerating sequence can exist. Kac and Ulam reason differently [20, p. 12-13]. They prove Theorem 1 and then Corollary 2. By combining these results, they obtain a non-constructive proof of the existence of transcendentals.
So Cantor does give a method of constructing transcendental numbers. By the way, Kac and Ulam present the same non-constructive proof that Stewart uses. — Preceding unsigned comment added by RJGray (talk • contribs) 01:56, 14 September 2018 (UTC)

OK, well, I haven't read Stewart's article; I don't know exactly what Stewart claims.  I was making an inference based on what the citation said.
If Stewart does not claim that Cantor didn't find a particular transcendental, then there's even less support for the claim of "disagreement". --Trovatore (talk) 06:06, 14 September 2018 (UTC)
If this doesn't qualify as actual disagreement, then maybe the DYK hook should get rephrased. But if it's not actual disagreement, it may still be accurate to say many actual mathematicians have misunderstood the matter in published writings. Many mathematicians including Dirichlet (who may be the originator of the error) have written in books and papers that Euclid's proof of the infinitude of primes is by contradiction. Most of that may be just following what they've read rather than substantially disagreeing. But some have written, erroneously, that Euclid's proof is non-constructive. "Disagreement"? Or (moderately) widespread error? If the latter, it could be a DYK hook if sufficiently supported. Michael Hardy (talk) 22:05, 14 September 2018 (UTC)
Trovatore, concerning your statement: 

If Stewart does not claim that Cantor didn't find a particular transcendental, then there's even less support for the claim of "disagreement".
Stewart said:

Meanwhile Georg Cantor, in 1874, had produced a revolutionary proof of the existence of transcendental numbers, without actually constructing any. 
I believe that most (nearly all?) people reading without actually constructing any would come away believing that Cantor's proof is a pure existence proof that does not construct transcendentals.
Cantor's constructive proof allows him to construct a transcendental number on any closed interval that is provided. Cantor was writing a research article and could expect that his readers would see this without him providing an example. In fact, he wrote at a time when non-constructive proofs were rare (this started to change in 1890 when Hilbert gave a non-constructive proof of his Basis Theorem). As pointed out in the Wikipedia article, even Kronecker would accept that Cantor's construction applied to the sequence of algebraic numbers produces a definite real number. This may be one reason why Cantor's article was accepted so quickly—only four days after submission. His next article suffered a long delay which he blamed on Kronecker. 
By the way, you bring up an important point—would intuitionists accept Cantor's proof? I believe the answer is no because his proof of his Theorem 2 uses the fact that an increasing or decreasing bounded sequence of reals has a limit. Intuitionists like limiting procedures that are coming from below and above. However, the footnote on page 821 in Georg Cantor and Transcendental Numbers says that page 27 in Bishop and Bridges Constructive Analysis (1985) has a proof of Theorem 2 that meets the demands of constructive mathematicians (and probably also intuitionists).
Concerning the word "disagree", I chose it to replace the word "controversy" in my first rewrite of the Wikipedia article. The word "controversy" has two problems: It's a "peacock" term and it's inaccurate because controversy implies that the mathematicians who are stating the proof is constructive or non-constructive are aware of the choice they are making. Disagreement simply means that what an article or book says disagrees with at least one source in the literature. I'm not particularly attached to the word "disagree". However, changing it could take some time (depending of course on the particular term chosen) and the GA review had no trouble with "disagree". 
Most sources seem to say that Cantor's proof is non-constructive. Ivor Grattan-Guinness in his two-sentence 1995 review of "Georg Cantor and Transcendental Numbers" stated: "It is commonly believed that Cantor's proof of the existence of transcendental numbers, published in 1874, merely proves an existence theorem. The author refutes this view by using a computer program to determine such a number". Working on this Cantor article rewrite, I still found more sources stating his proof is non-constructive, but it was nice to find a few saying the proof is constructive.
As far as the DYK hook: Michael, I think you are working on this. I'd be happy to help. Just give me your latest and best ideas.
Thank you, Michael and Trovatore, for the work you are doing on this. Now I have to work on adding a few references requested by the DYK review. --RJGray (talk) 19:02, 15 September 2018 (UTC)

RJGray:  There is not a disagreement.  There is not a controversy.  It's such a simple question that everyone agrees.  They just phrase it differently. --Trovatore (talk) 23:20, 1 May 2020 (UTC)

The lemma required for the second case in the proof states that x(n+1) and x(n+2) are the end points for the interval (a(n+1), b(n+1)), however these are just the first 2 candidates as end points, we do not know if they lie inside or outside the interval (a(n), b(n)). This is indeed enough to satisfy the criterion that x(n+1) and x(n+2) are either larger or best case are indeed the end-points.
The simplest modification would be to state that these two points are at best the end points. If not the induction actually runs better. 2001:1C02:1203:8500:49CD:D219:D882:A34 (talk) 13:32, 26 January 2023 (UTC)

"...he second column lists the terms visited during the search for the first two terms..."
what do you mean by "terms _visited_"? 217.149.171.204 (talk) 08:12, 5 July 2023 (UTC)