Talk:Cantor's first set theory article

Requested move 13 February 2016

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: No move. This RM has been open for nearly two months, and it seems clear that there isn't consensus for the move. Cúchullain ^t/c 15:11, 4 April 2016 (UTC)

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Georg Cantor's first set theory article → On a Property of the Collection of All Real Algebraic Numbers – Article is about this paper, so name it after the paper. See more elaborate remarks at the section #Title_containing_.22article.22 above. Trovatore (talk) 20:20, 13 February 2016 (UTC)

A few points to think about:

• The title of Cantor's article "On a Property of the Collection of All Real Algebraic Numbers" does not capture what the article is famous for--namely, the uncountability of the set of real numbers. Gödel's article "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I" does capture what the article is famous for. The reason for Cantor's choice of name is covered in the Wikipedia article.
• If we drop the "Georg" in the Wikipedia article's title, a reader who is interested in Cantor can type "Cantor" into the Search box and see the Wikipedia article along with other articles about Cantor's work. Being a reader who uses this feature of the Search box when I'm curious about a mathematician's or scientist's work, I think "Cantor's first set theory article" would be more reader-friendly than "On a Property of the Collection of All Real Algebraic Numbers."
• As far as "article" vs. "paper". See Difference Between Research Article and Research Paper. Two items of interest: "• There is a trend to refer to term papers and academic papers written by students in colleges as research papers whereas articles submitted by scholars and scientists with their groundbreaking research are termed as research articles. • Research articles are published in renowned scientific journals whereas papers written by students do not go to journals."--RJGray (talk) 21:35, 13 February 2016 (UTC)

1. As to your first point: Yes, that's true, but that's an interesting story in itself, which as far as I can see (but I haven't read the article carefully yet), the article does not currently discuss, but should probably be added. As I'm sure you know, Cantor is thought to have chosen this (somewhat inferior) title deliberately, to avoid a confrontation with Kronecker. Dauben goes into some detail on this point, I believe (don't have him to hand unfortunately). In any case I don't see that this should determine what this article should be called. This article is about the paper, and the paper has that name, and I really think that ought to settle it. Update — it looks like you say that the point about the reason for the title of Cantor's paper is covered in the Wikipedia article. I took a quick look, and I still don't see it; can you point me to it? I would expect it to be more prominent.
2. As to typing in the search box, it works for redirects too. Try it. There is in any case going to be a redirect from the current title, so I don't really see a problem.
3. I still strongly prefer "paper". To me a "paper" is more academic; an "article" is more likely to be for mass consumption. Also a "paper" is more likely to be a primary source, whereas an "article" is probably a secondary or tertiary source. Also the interference issue is real; the word "article" invites confusion with Wikipedia articles, whereas "paper" does not. But in any case we don't need to decide that in this RM as the proposed title does not contain either word. --Trovatore (talk) 21:46, 13 February 2016 (UTC)

I've looked more into "article" vs. "paper" and am I now neutral on the issue. But we can deal with this later. I do think it would be good to consult the readers to see if they think that a general change of "article" to "paper" throughout the entire Wikipedia article is a good idea. If they do, I'd be happy to make the change.

Let's talk about Wikipedia:Article titles and how your title compares to the existing title. Wikipedia says that: A good Wikipedia article title has the five following characteristics:

• Recognizability – The title is a name or description of the subject that someone familiar with, although not necessarily an expert in, the subject area will recognize. Your suggested title On a Property of the Collection of All Real Algebraic Numbers is unfortunately not recognizable even to students taking set theory unless they have read a historical work that discusses the reason for this strange name. Georg Cantor's first set theory article is recognizable because it's talking about Cantor's work and, in particular, his first article on set theory.
• Naturalness – The title is one that readers are likely to look or search for and that editors would naturally use to link to the article from other articles. Such a title usually conveys what the subject is actually called in English. I don't think that readers will look or search for On a Property of the Collection of All Real Algebraic Numbers while they will search for Cantor's first set theory article since they are likely to start typing "Cantor" and unlikely to start typing "On a Property." Also, since Cantor's first set theory article is a natural shortening of the current title, there is no need to boldface Cantor's first set theory article in the text.
• Precision – The title unambiguously identifies the article's subject and distinguishes it from other subjects. On a Property of the Collection of All Real Algebraic Numbers does not even identify the property that is discussed in Cantor's article. It certainly doesn't capture the Wikipedia article's subject.
• Conciseness – The title is no longer than necessary to identify the article's subject and distinguish it from other subjects. Georg Cantor's first set theory article is shorter than On a Property of the Collection of All Real Algebraic Numbers
• Consistency – The title is consistent with the pattern of similar articles' titles. There are other titles with "Cantor's ..." in it, while your title shares consistency with the Wikipedia article on the Gödel paper.

I also decided to see what effect your suggested title and a redirect would have on the lead. Here's the modified lead (I removed the refs):

On a Property of the Collection of All Real Algebraic Numbers is Georg Cantor's first set theory article. It was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the paper, "On a Property of the Collection of All Real Algebraic Numbers," refers to its first theorem: the set of real algebraic numbers is countable.

One problem with your suggested title is that readers who are redirected from Cantor's first uncountability proof may get confused when redirected to a Wikipedia article titled On a Property of the Collection of All Real Algebraic Numbers. The current redirect to Georg Cantor's first set theory article is less confusing because of the "Cantor's first" in the title and because the 2nd sentence in the current article talks about uncountably infinite and the next sentence has Cantor's first uncountability proof in it.

Also, the first sentence in the modified lead is only necessary because of the suggested title change, which also requires the boldfaced Georg Cantor's first set theory article to handle the redirect. I believe quicker leads are better because they entice users to read the article. Also, I wrote the original lead to relegate the obscure title of Cantor's paper to the bottom of the paragraph since the Wikipedia article doesn't devote much space on the countability of the real algebraic numbers. (I deal with the title more in the section "The influence of Weierstrass and Kronecker on Cantor's article.")

I guess my feeling is Cantor got stuck with a poor title for his paper. I don't think we need to get stuck with the same obscure title.--RJGray (talk) 00:43, 15 February 2016 (UTC)

The article is about a published work, so we go with the title of that work. I don't know any exception to that. If you had made it about the content, then there would be more options, but you made it about the paper itself, so I think there is really only one choice. --Trovatore (talk) 05:37, 15 February 2016 (UTC)

I am inclined to keep the present title. @Trovatore: Have you attempted to find out which articles on published works exist on Wikipedia and examine them to see whether some Wikipedia article titles differ from the titles of the published works? Michael Hardy (talk) 01:40, 16 February 2016 (UTC)

I'm not sure how I would really do that except by serially looking through articles on published works. I'm not aware of any exception.

Can we agree that, as article titles, descriptions are inferior to names, assuming a canonical name exists? I would think that's kind of obvious, actually. Sometimes there is no agreed name, and you have to fall back to a description, but that's an unfortunate necessity. But pretty much every published work has a name, namely its title, so I don't see any justification for titling this article with a description. --Trovatore (talk) 19:17, 16 February 2016 (UTC)

On suggested move:

On Formally Undecidable Propositions of Principia Mathematica and Related Systems

This article is about the paper. For theorems proved in this paper, see Gödel's incompleteness theorems.

I'm not sure what you're trying to say here. Are you saying there's a disanalogy because this article is (currently) more about the content than about the paper per se? If so, then the title should refer to the content, and not to the published work; it currently refers to the published work.

But really I don't think the content of the paper is a very natural topic for an article, given the divergent character of the two results. I think we should have an article about the paper, and I think it should be named after the paper, and it should spend more time on the paper per se than it currently does. (For example, currently, the article doesn't even seem to give the journal in which the paper was published, which I believe was Crelle's Journal.) --Trovatore (talk) 21:59, 16 February 2016 (UTC)

Sorry, I thought I was writing in one of my private files. I didn't realize until later that I wasn't. But you got part of the gist of what I planned to write about. I'm thinking that there should be two Wikipedia articles, similar to the way Gödel's work is divided into two parts. The article on Gödel's paper just covers material on the paper itself. I'd be happy to write a second article that would be similar in format to On Formally Undecidable Propositions of Principia Mathematica and Related Systems and would be called On a Property of the Collection of All Real Algebraic Numbers. It would include an outline of the Cantor's article, which by the way also contains an improvement of a theorem of Minnigerode that most coverage of the article leaves out (Dauben does mention it). It would also cover translations of the article, including the 1883 French translation and how that came about (and, of course, Ewald's English translation). It would also give the German name of the article and the journal in which it was published. By the way, in the current article, the section Georg Cantor's first set theory article#The influence of Weierstrass and Kronecker on Cantor's article contains the sentence: "Cantor would submit his article to Crelle's Journal."

I think one problem we are facing is that the current Wikipedia article goes far beyond Cantor's article: Starting with the Development section, it mentions 16 other mathematicians. So titling the article On a Property of the Collection of All Real Algebraic Numbers doesn't capture the article's content but would capture the content of the 2nd article I'm offering to write. The math history section of the current Wikipedia article covers a slice of math history that came about because of Cantor's article. For everyone reading this: I welcome suggestions as what to call the current Wikipedia article that would accurately capture both its math and math history content (I'm working on this myself). In the absence of title suggestions: What's wrong with having two articles, one titled On a Property of the Collection of All Real Algebraic Numbers and one titled Georg Cantor's first set theory article? RJGray (talk) 22:50, 16 February 2016 (UTC)

I believe strongly that a Wikipedia article should be about one clear thing, and that thing should be named in the title. If the title refers to the paper, then the article should be about the paper, and I think should be named after the paper. (We wouldn't put the article about Gone With the Wind at Margaret Mitchell's most famous novel, say.)

You seem to be saying this article should be about the content of the paper. I don't think that's a natural topic for an article, the two results being so divergent. I think the article is fine, but it should say more about the paper per se, and be named after the paper.

I did find the reference to Crelle's Journal, later, but it's too buried. If the article is about the paper, which I think it should be, then the journal and publication date should be named in the first paragraph, probably in the first sentence. --Trovatore (talk) 22:59, 16 February 2016 (UTC)

I, for one, support the move. As WP:TITLE says, "In Wikipedia, an article title is a natural language word or expression that indicates the subject of the article: as such the article title is usually the name of the person, or of the place, or of whatever else the topic of the article is." If the topic of the present article is Georg Cantor's first set theory article, then the article title almost surely should be the title of that article.

If that doesn't capture the scope of the article, then the article should be revised in one way or another. We can either abandon the idea that the article is about one of Cantor's papers, or we can move some of the material less relevant to Cantor's paper to, say, History of set theory. I don't see any need for the latter article to spring fully-formed from our foreheads; it could be started and left incomplete for now. Or, if the result would be too short, then I think it's fine for the present article to overgrow its proper scope. Eventually the relevant material can be reorganized. Ozob (talk) 00:25, 17 February 2016 (UTC)

Several more points:

• I can find no explicit Wikipedia rule that requires the name of an article be used. In fact, I've already discovered three examples where it is not used:
  • Wikipedia article title: Grothendieck's Tôhoku paper. Paper's name: "Sur quelques points d'algèbre homologique." (Mathematics)
  • Wikipedia article title: Alpher–Bethe–Gamow paper. Paper's name: "The Origin of Chemical Elements." (Physics)
  • Wikipedia article title: Lighthill report. Paper's name: "Artificial Intelligence: A General Survey." (Computer Science)
• WP:TITLE states: "A good Wikipedia article title has the five following characteristics: Recognizability … Naturalness … Precision … Conciseness … Consistency. I discussed these above and compared the current Wikipedia article title with the proposal to use the title of Cantor's article. No one has refuted my claim that the current title is better than the proposed title in the first four characteristics and in the last characteristic they are tied.
• Concerning the example: "We wouldn't put the article about Gone With the Wind at Margaret Mitchell's most famous novel, say." For me, Margaret Mitchell's most famous novel doesn't work because of Recognizability. I've seen the movie Gone With the Wind so I would recognize it. However, I've never read the book so I wouldn't recognize the author's name. In the case of On a Property of the Collection of All Real Algebraic Numbers: many (most?) mathematicians and students of mathematics would not recognize this title, but nearly all of them would recognize the author's name in Georg Cantor's first set theory article. They would also recognize the area of "set theory" and a good number could tell you his first article's most significant result: the uncountability of the set of real numbers. These are two of the problems with the proposed title: (1) It fails the WP:TITLE Recognizability characteristic: "The title is a name or description of the subject that someone familiar with, although not necessarily an expert in, the subject area will recognize." (2) The title is unexpected and confusing since it doesn't mention the theorem the article is well-known for. (By the way, Margaret Mitchell's most famous novel is not an acceptable Wikipedia title because it uses the peacock word "famous"--see WP:PEACOCK).
• Concerning Wikipedia article content: I regard there as being two approaches to article content: A narrow article that only talks about what is in the math article versus a more comprehensive article that does this and puts it into historical context. In mathematics and the sciences, it is often of interest to understand what led to an article and what an article has led to. In the case of Gödel's article, the Wikipedia article "Gödel's incompleteness theorems" is the comprehensive article and On Formally Undecidable Propositions of Principia Mathematica and Related Systems is the narrow article (it only contains publication info, outline of the paper, and a section on translations of the paper). I was thinking of having the same division with current Wikipedia article title being the more comprehensive article and another Wikipedia article (whose title would be the title of Cantor's article) that would be very narrow. It was a compromise measure I was proposing, but it obviously got nowhere and I only confused people. I take it that everyone wants just one Wikipedia article (which I think is the best way to go). --RJGray (talk) 17:12, 19 February 2016 (UTC)

I really don't think there's a choice here. If the article is about the paper, which I think it should be, then it should be named after the paper. --Trovatore (talk) 17:49, 19 February 2016 (UTC)

I looked back and saw your examples. That is a point. Still, no one treats "Cantor's first set theory article" as a name of the paper (whereas for Alpher-Beta-Gamow they arguably do). The current title is a description; that's what offends me the most about it. Descriptions are the last choice for WP article titles. --Trovatore (talk) 17:52, 19 February 2016 (UTC)

A better example than the Alpher-Beta-Gamow paper is the Grothendieck paper. The title Grothendieck's Tôhoku paper is as descriptive as Cantor's first set theory article. The former is saying the Wikipedia article is about Grothendieck's paper that appeared in the Tôhoku Mathematical Journal, while the latter is saying the Wikipedia article is about Cantor's first article on set theory.

I think that descriptions for WP article titles should only be used if there are good reasons for their use. Usually, I would avoid using a description for a WP article title. However, as I pointed out above we are in an unusual case where Cantor was coaxed (or pressured) into choosing a title that has nothing to do with the revolutionary result the article is famous for. Hence, the descriptive title easily beats the title of Cantor's article as measured by the five characteristics of a good Wikipedia article title (WP:TITLE). We also need to think of readers who come to the Wikipedia article via various links; for example, at least 20 articles have links to "Cantor's first uncountability proof." Not all these readers will be well-versed in math since some of the links are in an easier section of an article. These readers will find themselves at an article with a long name — "On a Property of the Collection of All Real Algebraic Numbers" — that's talking about some unspecified property of the collection of real algebraic numbers. I don't find this reader-friendly at all and I wouldn't be surprised if some readers think they're not at the correct article and just leave.

Also, I like to think of the first sentence of an article as a welcome mat. Starting an article with "Georg Cantor's first set theory article" is far more welcoming.

Finally, some books completely avoid the name of Cantor's paper and use a description similar to the current Wikipedia article title. Here's a couple of examples: Hao Wang, Popular Lectures on Mathematical Logic, p. 119: "Cantor's first paper on set theory was published in 1874." Gerard Buskes and Arnoud van Rooij, Topological Spaces: From Distance to Neighborhood, page 112: "Cantor's first set theory paper was his 1874 paper on algebraic numbers." It seems that they think the article is about algebraic numbers (perhaps the title confused them), but they do go on to cover the article's theorem about the uncountability of the real numbers.

I do like the respectful tone of our discussion and it has challenged me to think about why I have the opinion I do. Perhaps the next discussion we're both involved in will have us in agreement. RJGray (talk) 02:56, 20 February 2016 (UTC)

The article Grothendieck's Tôhoku paper has the advantage that people do in fact refer to that paper as "Grothendieck's Tôhoku paper" both in conversation and in writing. I don't ever recall having a conversation with anyone in which Cantor's first set theory article was discussed (but maybe that's because I work in algebraic geometry and not set theory or logic) so I don't know how people refer to it.

Neither the present title nor "Grothendieck's Tôhoku paper" strike me as being appropriate. After all, the papers have names, and I continue to recoil at the idea of calling them something other than their names. Though I'm reminded of Haddocks' Eyes. Ozob (talk) 04:28, 20 February 2016 (UTC)

• Oppose this complicated proposal. The present title seems fine. Dicklyon (talk) 05:07, 4 April 2016 (UTC)

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The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

A replacement for the removed sentence

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)

Re-evaluating refs in lead

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)

Problems with nowrap and sfrac templates

In the following sentence, by shrinking your window, in the first use of nowrap, you can get the subscript 1 in "a_n1" or one of the other subscripts to wrap to the next line. In the second use of nowrap, you can get the comma in "n_ν," to wrap to the next line. Is there anywhere we can submit a bug report on nowrap? I switched over to " ".

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 (a_{n1, n2, …, nν}) where n₁, n₂, …, n_ν, 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, a_{n1, 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  . I think that is similarly broken. —David Eppstein (talk) 18:50, 4 May 2016 (UTC)

Actually, I've got them working with   except in the one case of a subscript 2n – 1 where I had to use a nowrap within the _… . 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)

Problems with " " in Internet Explorer and with "nowrap" in Chrome

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 " " 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(t_{2n – 1}) = t_n, and g(t_2n) = a_n.

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(t_{2n – 1}) = t_n, and g(t_2n) = a_n.

I've just fixed the 1st occurrence so that at least it works in both browsers. In many cases " " 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 " " 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)

Collapsed content

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)

Reply to "Why doesn't Cantor's second theorem apply to a world of purely algebraic numbers?"

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 (a_N, b_N) 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 ${\sqrt {2}}-1$, 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)

"Disagreement" on constructivity

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, ..., ω_η, ... 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)

Proof of Lemma needs a change

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)

visited?

"...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)