Talk:Cantor's first set theory article: Difference between revisions

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|dykentry=... that mathematicians disagree about whether a proof in [[Georg Cantor]]'s '''[[Georg Cantor's first set theory article|first set theory article]]''' actually shows how to construct a [[transcendental number]], or merely proves that such numbers exist?

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== Title containing "article" ==

This looks like a fascinating (Wikipedia) article, and I'm looking forward to reading it in detail.

I'm not too convinced by the title, though. I think it's more usual to refer to such contributions as "papers" rather than "articles". To me "article" sounds like something you find in a magazine, not a journal. Also, as I alluded to above, it's a bit problematic that "article" also is used to refer to Wikipedia articles, and there's possible interference from that, for editors discussing the Wikipedia article, but more importantly also for readers.

We ''could'' move it to [[Georg Cantor's first set theory paper]], but to be honest I would rather move it to the actual title, ''[[On a Property of the Collection of All Real Algebraic Numbers]]'', and put it in italics. I think in general we write articles on notable papers by their titles. See for example [[On Formally Undecidable Propositions of Principia Mathematica and Related Systems]] (not sure why it's not in italics; I think it should be). --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 03:38, 13 February 2016 (UTC)

:<small>Just an update on the italics issue &mdash; I went ahead and added {{tl|italic title}} to the other article. But now I'm having second thoughts; there may be an argument for preserving a distinction between book titles, which are italicized, and titles of papers, which are not. I don't know. I think it looks better with italics; it makes it clearer that it's a title of a published work. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 03:59, 13 February 2016 (UTC) </small>

== The article rewrite and thanks to all those who helped me ==

It's been challenging rewriting the "Cantor's first uncountability proof" article because it's listed in the categories: History of mathematics, Set theory, Real analysis, Georg Cantor. So I had to consider both the math and the math history audiences. I did this by writing the article so all the math in Cantor's article appears in the first two sections, which is followed by a "Development" section that acts as a bridge from the math sections to the math history sections. I changed the title to "Cantor's first set theory article" to reflect its content better; actually, the old article could have used this title. It's a well-known, often-cited, and much-discussed article so I suspect the Wikipedia article will attract a number of readers.

I would like to thank SpinningSpark for his excellent critique of the old article. I really appreciate the time and thought he put into it. The new article owes a lot to SpinningSpark. His detailed section-by-section list of flaws was extremely helpful. I used this list and his suggestions to restructure and rewrite the article. I particularly liked his comment on whether the disagreement about Cantor's proof of the existence of transcendentals "has been a decades long dispute with neither side ever realising that they were not talking about the same proof." The lead now points out this disagreement has been around at least since 1930 and still seems to be unresolved. It was a major flaw of the old article that the longevity of the disagreement was never mentioned. I find it ironic that this disagreement is still around, while most mathematicians now accept transfinite (infinite) sets so the old dispute about the validity of these sets is mostly resolved.

I also thank JohnBlackburn for his comments. His comments that made me realize that I should think of the readers who just want to understand the math in Cantor's proof. This led to the restructuring mentioned above in the first paragraph. His comments also led me to put the long footnotes containing math proofs into the text. I also added some more math to the article.

I thank Jochen Burghardt for his help on the rewrite. He did the case diagrams for the proof of Cantor's second theorem, the subsectioning of "The Proof" section, the calculations in the table "Cantor's enumeration of the real algebraic numbers", and he pointed out places where my writing was unclear. The need for the case diagrams came from reading SpinningSpark's comment on what is now Case 1. I realized that a reader's possible confusion on whether there is point in the finite interval (''a_N'', ''b_N'') besides ''x_n'' could be handled with a diagram. I contacted Jochen with three simple ASCII diagrams. He took my simplistic diagrams and produced diagrams that capture the dynamics of the limiting process.

I thank my daughter Kristen who read a recent draft and made a number of suggestions that improved the writing. Especially important were her suggestions on improving the lead.

I also thank those who edited the old article. I started with a copy of the old article and have kept up with recent edits so that your edits would be preserved (except perhaps in the parts of the article where large changes were made).

Finally, I wish to thank Michael Hardy for his GA nomination for the old article, for giving me the go-ahead for the rewrite, and for his patience with the amount of time it has taken me to do the rewrite. I hope that this rewrite is much closer to GA standards than the old article. --[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 15:26, 13 February 2016 (UTC)

== No need to merge article with Cantor's first uncountability proof -- already contains its content ==

There is no need to merge this article with [[Cantor's first uncountability proof]] because this is a rewrite of that article with guidance from the GA Review of that article. What is needed is a redirect from [[Cantor's first uncountability proof]] to this new article. Note that '''Cantor's first uncountability proof''' appears in boldface in first paragraph so readers will know that this article contains the content of that article. --[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 15:37, 13 February 2016 (UTC)

:Ok, but in that case we needed to merge the histories to make it clearer that much of the content was rewritten from the older version. I have completed a history merge, so now the histories of both articles are in one place, and removed the merge tags. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 21:41, 13 February 2016 (UTC)

== Requested move 13 February 2016 ==

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* '''Oppose''' this complicated proposal. The present title seems fine. [[User:Dicklyon|Dicklyon]] ([[User talk:Dicklyon|talk]]) 05:07, 4 April 2016 (UTC)

----

:''The above discussion is preserved as an archive of a [[Wikipedia:Requested moves|requested move]]. <span style="color:red">'''Please do not modify it.'''</span> Subsequent comments should be made in a new section on this talk page or in a [[Wikipedia:Move review|move review]]. No further edits should be made to this section.''</div><!-- Template:RM bottom -->

== Constructive? ==

The lede text on constructiveness seems a bit confused. There's not clear evidence for a controversy; the only "against" is "Stewart, 2015" but that's not enough to identify who Stewart is. Or who Sheppard 2014 might be. Fraenkel is a heavyweight though so you'd need a good reason to disagree with him [[User:William M. Connolley|William M. Connolley]] ([[User talk:William M. Connolley|talk]]) 22:46, 2 April 2016 (UTC)

:Those are standard Harvard citations and the sources are listed at the end, so it is clear enough to me what the references point to. There seems to be an entire section of the article related to this issue, with numerous sources. &mdash;&nbsp;Carl <small>([[User:CBM|CBM]]&nbsp;·&nbsp;[[User talk:CBM|talk]])</small> 15:05, 3 April 2016 (UTC)

== A replacement for the removed sentence ==

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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: <nowiki> {{browser | Chrome | … }} {{browser | Internet Explorer | … }} {{browser | default | … }} </nowiki>. Of course, it would be preferable for this to be done by the Wikipedia people who maintain "&amp;nbsp;" and "nowrap". Can my examples be communicated to them? Thanks, [[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 14:31, 6 May 2016 (UTC)

:: {{ping|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. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 21:14, 14 September 2018 (UTC)

== Collapsed content ==

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{{Did you know nominations/Georg Cantor's first set theory article}}

== 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'',&nbsp;''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'',&nbsp;''b''_''N'') or [''a''_∞,&nbsp;''b''_∞], respectively. However, in case 2 where ''a''_∞&nbsp;=&nbsp;''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 <math>\sqrt{2}-1</math>, 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'',&nbsp;''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. --[[User:RJGray|RJGray]] ([[User talk: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.) --[[User:Trovatore|Trovatore]] ([[User talk: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 [http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf 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. <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:RJGray|RJGray]] ([[User talk:RJGray#top|talk]] • [[Special:Contributions/RJGray|contribs]]) 01:56, 14 September 2018 (UTC)</small>

::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". --[[User:Trovatore|Trovatore]] ([[User talk: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. [[User:Michael Hardy|Michael Hardy]] ([[User talk: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 [http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf 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. --[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 19:02, 15 September 2018 (UTC)

:{{U|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. --[[User:Trovatore|Trovatore]] ([[User talk: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. [[Special:Contributions/2001:1C02:1203:8500:49CD:D219:D882:A34|2001:1C02:1203:8500:49CD:D219:D882:A34]] ([[User talk: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_"? [[Special:Contributions/217.149.171.204|217.149.171.204]] ([[User talk:217.149.171.204|talk]]) 08:12, 5 July 2023 (UTC)