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

Revision as of 22:05, 14 September 2018

Revision as of 19:02, 15 September 2018

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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 "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)