Talk:Cantor's first set theory article/GA1

GA Review

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Reviewer: Spinningspark (talk · contribs) 21:31, 3 December 2014 (UTC)[reply]

Hi Michael, I just saw this fascinating subject in passing, so I 'm going to review it, even though I don't usually review maths articles. I'm going to be busy tomorrow so might not be able to do a full review for a few days, but one thing jumps out at me straight away so I'll mention that now.

Is there really a controversy over the constructiveness of the proof, or is it merely two groups of mathematicians talking at cross purposes not understanding that the other is talking about a different proof? That would seem to be the case, but is far from clear in the lead. It strikes me that undue emphasis is being given to this when it amounts only to a mere misunderstanding of the other. Unless of course there really has been a decades long dispute with neither side ever realising that they were not talking about the same proof.

One minor point, MOS:HEAD says that headings should not contain questions. Although that is not actually something covered by the GA criteria. SpinningSpark 21:31, 3 December 2014 (UTC)[reply]

Review

I have already raised the issue of whether the so-called controversies discussed in the article are genuinely controversial points. Looking through a few previews in gbooks, it is obvious that Cantor's work was indeed controversial, but more on the question of whether it was valid to study transfinite sets at all rather than the issues raised in our article. I may be completely wrong about this (not my subject) but that's the way it comes across to me. If so, there is a big hole in the article coverage which is a failure of GA criterion 3a.

Lead

See my initial comment on the nature of the controversy
Wikilinking part of the bolded part of the lead sentence is discouraged by WP:LEAD.
Leaving two unanswered questions is not a sufficient summary of the article content on them
There is a lot in the article about the history of the idea (whole section on development) but absent from the lead.

The article

"Note how Cantor's second theorem..." There is a WP:WTW issue here.
"...Cantor's second theorem separates the constructive content of his work from the proof by contradiction..." I think it needs stating explicitly that the second theorem is the constructive content (if indeed that is the case)
Transcendental number should be wikilinked. It is linked later in the article, but that is not the first mention.
The paragraph beginning "The first half of this remark..." is uncited. At the very least it needs a cite for "Cantor...probably" which is making a theory of mind statement about Cantor's motives.

The proofs

"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.
"...thus indicating how his proof handles this particular sequence". Not particulary clear what this is saying.

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

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.
"... or it uses his diagonal method." If this is referring to Cantor's diagonal argument it should be wikilinked. The page is already wikilinked, but later in the article.
"The constructive nature of Cantor's work is easily demonstrated by using his two methods to construct irrational numbers." Apparently contradicting "one proof is constructive while the other is non-constructive".
Why are we suddenly discussing irrationals here? The dispute in question is over the constructibility of transcendentals, not irrationals.

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

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.

See also

Links already in the body of the article should not be repeated in see also.

Images

Likenesses of Cantor and other major mathematicians in this story are available. Why not use them in this article?

SpinningSpark 16:04, 6 December 2014 (UTC)[reply]

Hello. Thank you for working on this. I've done a few edits today and I'll be back. Michael Hardy (talk) 00:10, 10 December 2014 (UTC)[reply]

The review says:

"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. Michael Hardy (talk) 23:55, 19 December 2014 (UTC)[reply]

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). SpinningSpark 13:05, 20 December 2014 (UTC)[reply]

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. SpinningSpark 13:05, 20 December 2014 (UTC)[reply]