Talk:Cantor's first set theory article/GA1: Difference between revisions

Revision as of 23:55, 19 December 2014

Revision as of 13:06, 20 December 2014

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;The proofs

*[[File:Symbol confirmed.svg|16px]] "interior of the interval" requires a definition (in running text, by gloss, or by wikilink) as it has a specific technical meaning here.

*"Since at most one xn can belong to the interior of [aN, bN], any number belonging to the interior of [aN, bN] besides xn is not contained in the given sequence." There seems to be an implication here that one can prove that there is a number other than xn in the interior, or am I missing something? In any case, this part of the proof does not seem to have been brought to a conclusion.

*"Cantor observes that the sequence of real algebraic numbers falls into the first case..." There is an ambiguity here over which case is being discussed. There is a first case and second case of finite or infinite intervals, and then the second case has a first case and second case.

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;Is Cantor's proof of the existence of transcendentals constructive or non-constructive?

* [[File:Symbol confirmed.svg|16px]] Headings should not contain questions per MOS

* See my initial comment on the nature of the controversy

* One has to do a great deal of reading between the lines, or going back to earlier in the article to get the basics of which proof is being discussed here: which proof is the subject of ''this'' article, whether the 1874 proof is a synonym for the subject of this article, whether the subject of this article is a constructive proof or not, and whether mathematicians cited are discussing the subject of ''this'' page or not.

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;Why does Cantor's article emphasize the countability of the algebraic numbers?

* [[File:Symbol confirmed.svg|16px]] Question in heading

* "This has led to a controversy." This is uncited and seems to be an overstatement of what I can see in the article. Dauben says it was influenced by Kronecker and Ferreirós says it was influenced by both Kronecker and Weierstrass. Hardly a controversy, a slight difference in emphasis maybe.

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: "There seems to be an implication here that one can prove that there is a number other than xn in the interior, or am I missing something?".

That is correct: there must be such a number since there are infinitely many numbers in the interval. But it is not clear what you're suggesting should be done about it, as far as editing the article is concenrred. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 23:55, 19 December 2014 (UTC)

::That is what needs saying, since there are infinitely many numbers in any given finite interval there must be a number other than xn. The implication is there, but the article fails to explicitly say this is why it is proved. I don't think that step is going to be obvious to all readers. It is not even obvious that one is still left with a finite interval. (I am not disputing anything here of course, just looking at it from the perspective of someone completely unfamiliar with the material). [[User:Spinningspark|<b style="background:#FAFAD2;color:#C08000">Spinning</b>]][[User talk:Spinningspark|<b style="color:#4840A0">Spark</b>]] 13:05, 20 December 2014 (UTC)

I see that you have rephrased the subheadings so that they are no longer questions and tinkered with the corresponding phrasing in the lead. I am afraid this is not really getting to the heart of the matter. I think some structural changes to the article need to be made to take the emphasis off this alleged dispute/disagreement. The disagreement does not seem to amount to a whole pile of beans. If it does, some sources saying so are needed. Even more, fundamentally from a GA perspective (criterion 3b), the discussion of this dispute is part of a tendency for the article to go off at a tangent to discuss Cantor's other proof(s). The non-constructive proof is the diagonal argument, no? which is not the subject of this article. I have already commented on how easily the reader can become confused over which proof is being discussed. The diagonal argument should be discussed only inasmuch as it is needed to describe ''this'' method, or in passing to say Cantor went on to use other methods. [[User:Spinningspark|<b style="background:#FAFAD2;color:#C08000">Spinning</b>]][[User talk:Spinningspark|<b style="color:#4840A0">Spark</b>]] 13:05, 20 December 2014 (UTC)