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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== 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)

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:''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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::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)