Talk:Cantor's first set theory article/GA2

GA Review

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Reviewer: Bilorv (talk · contribs) 01:50, 29 July 2018 (UTC)

Having just finished my first year as an undergraduate, this article's technical content is slightly testing of my abilities but it's on such an interesting topic that I couldn't resist. — Bilorv_(c)(talk) 01:50, 29 July 2018 (UTC)

GA review (see here for what the criteria are, and here for what they are not)

1. It is reasonably well written.

   a (prose, spelling, and grammar): b (MoS for lead, layout, word choice, fiction, and lists):

2. It is factually accurate and verifiable.

   a (reference section): b (citations to reliable sources): c (OR): d (copyvio and plagiarism):

3. It is broad in its coverage.

   a (major aspects): b (focused):

4. It follows the neutral point of view policy.

   Fair representation without bias:

5. It is stable.

   No edit wars, etc.:

6. It is illustrated by images and other media, where possible and appropriate.

   a (images are tagged and non-free content have fair use rationales): b (appropriate use with suitable captions):

7. Overall:

   Pass/Fail:

The article

• "Cantor's article is short, not even four and a half pages" — Can we trim this to "Cantor's article is under five pages long" or "Cantor's article is roughly four and a half pages"?
• "It begins with a discussion of the real algebraic numbers" — Would "definition" be better than "discussion"?
• "Cantor restates this theorem in terms more familiar to mathematicians of his time" — This seems to be a stronger claim than is verified by the paper itself (or its English translation).
• "The first part of this theorem implies the "Hence" part ..." — Agreed, but Cantor doesn't describe this in the article, right? Or am I missing the point/context of this paragraph?

The proofs

• I'm confused about the sourcing for these subsections, other than the parts which are direct rephrasing of Cantor's article. In particular, it seems like there should be inline citations somewhere for the table in the first proof, the closed interval simplification in the second proof and the entirety of the example of Cantor's construction.

The development of Cantor's ideas

• "Liouville's theorem that there are transcendental numbers" — Is this the same theorem mentioned above that there exist infinitely many transcendental numbers in any closed interval? Or an earlier theorem of Liouville's? If it's the same theorem, the link probably belongs under its first mention (and rephrasing to connect the two mentions would be good); if not, ignore me.

The disagreement about Cantor's existence proof

• In the third paragraph, the "exact" asymptotics $O(2^{\sqrt[{3}]{n}})$ and $O(n^{2}\log ^{2}n\log \log n)$ might as well be mentioned.
• "The proof of Cantor's second theorem does not state why some limits exist. The proof he was using does." — I don't quite understand either of these sentences. My understanding from reading the paper is that Cantor implicitly invokes what I would describe as the Monotonic Sequence Theorem to deduce that $\alpha ^{\infty }$ and $\beta ^{\infty }$ exist. Does this bullet point mean that Cantor is using this without statement, but requires it for his claims? Or that Cantor didn't explain this bit but the editor added in an explanation? Or something completely different?
• Is the 1870 Heine article relevant to Cantor or his work (e.g. a close friend / publication on a similar topic)? Is the point here that Kronecker is delaying publication in the journal because of his finitist views? (If so, add a few words to spell this out.)
• "Cantor chose "On a Property of the Collection of All Real Algebraic Numbers,"" — I would put the German title here and then the English in parentheticals (pedantic, but the German name is what Cantor was really choosing).

The legacy of Cantor's article

• "In 1922, Thoralf Skolem proved that if the axioms of set theory are consistent" — Which axioms of set theory exactly? Is this referring to a specific collection of axioms (e.g. ZFC), or saying generally "given any set of consistent axioms ..."? (In the latter case, the definite article "the axioms of set theory" is misleading.)
• "this does not contradict Cantor's uncountability theorem" is an Easter egg link; mention the phrase "Skolem's paradox" in prose.

Lead

• Add a link to Georg Cantor somewhere (but not in the bolded text). In fact, it's not really necessary to use the phrase "Georg Cantor's first set theory article" verbatim in the first sentence: I recommend "Georg Cantor published his first set theory article in 1874; it contains ..."
• "One of these theorems is "Cantor's revolutionary discovery" that ..." — It's not clear where this quote originates, and it's also not mentioned in the body of the article.
• "Cantor's article also contains a proof of the existence of transcendental numbers" — I would mention their infinitude as well.
• "In addition, they have looked at the article's legacy" — I think it would be better to actually describe the legacy e.g. "The uncountability theorem and the concept of countability have had a significant impact on mathematics, and Cantor's later work followed on from these ideas."

I'm putting the article formally  On hold now, so you have seven days to begin addressing the points above. — Bilorv_(c)(talk) 01:50, 29 July 2018 (UTC)

Comments

• The review says this:

  "Cantor's article is short, not even four and a half pages" — Can we trim this to "Cantor's article is under five pages long" or "Cantor's article is roughly four and a half pages"?

  But I think it is better as it is. Informing the reader that that length is small by comparison to what was (and is) typical is relevant.

But that's an unreferenced claim. The classic Synthesis example is about "160" vs "only 160", and it's a similar thing here. Just say "four and a half pages", and the readers can easily work out for themselves that that's quite short. — Bilorv_(c)(talk) 18:40, 4 August 2018 (UTC)

Then the review says

"It begins with a discussion of the real algebraic numbers" — Would "definition" be better than "discussion"?

There is indeed a terse definition at the beginning. Then there is a longer discussion that includes such things as the fact that every interval about a real algebraic number contains infinitely many others.

Yes, that makes sense. — Bilorv_(c)(talk) 18:40, 4 August 2018 (UTC)

Next the review says

"The first part of this theorem implies the "Hence" part ..." — Agreed, but Cantor doesn't describe this in the article, right? Or am I missing the point/context of this paragraph?

But should we have no discussion of trivial consequences of what Cantor said?

This initially confused me because I thought the section was solely about points made in the article, but I now see that "Cantor observes" and similar are used to distinguish points made by Cantor and secondary commentary. My mistake. I would prefer a source for the paragraph but I'm happy to let it go, on account of Uncontroversial knowledge. — Bilorv_(c)(talk) 18:40, 4 August 2018 (UTC)

Then we see this:

it seems like there should be inline citations somewhere for the table in the first proof, the closed interval simplification in the second proof and the entirety of the example of Cantor's construction

@Bilorv:What do you mean by "the closed interval simplification"?

Sorry for the weird phrasing. I was referring to: "We simplify Cantor's proof by using open intervals." — Bilorv_(c)(talk) 18:40, 4 August 2018 (UTC)

Next:

"One of these theorems is "Cantor's revolutionary discovery" that ..." — It's not clear where this quote originates

But a work of Joseph Dauben is cited.

My mistake—I thought that citation covered the uncountability of the reals but not the quote. — Bilorv_(c)(talk) 18:40, 4 August 2018 (UTC)

Michael Hardy (talk) 18:11, 4 August 2018 (UTC)

(So you're aware, I've changed this to a level 3 header per the "Please add all review comments..." hidden note at the top of the page. I'm also interspersing replies because I think that's simpler – feel free to move them if you prefer, or move your comments above to intersperse with my original review.) — Bilorv_(c)(talk) 18:40, 4 August 2018 (UTC)

Rewrote sentence to eliminate Easter egg link to Skolem's paradox. —RJGray (talk) 20:41, 5 August 2018 (UTC)

Concerning "infinitude of transcendentals" in Lead. In WP:Lead, it states: "The lead serves as an introduction to the article and a summary of its most important contents." The most important content is that Cantor devised a new method of proving the existence of transcendentals. Their infinitude follows easily from their existence in many different ways—one simple way is by adding rationals to a transcendental, which has nothing to do with Cantor's work. The fact that Cantor's method can also generate infinitely many transcendentals is just one of many ways to prove their infinitude and is a detail that I feel is best left to the later discussion of his method. —RJGray (talk) 21:02, 5 August 2018 (UTC)

Okay, fair enough. — Bilorv_(c)(talk) 22:54, 5 August 2018 (UTC)

Concerning "Cantor chose 'On a Property of the Collection of All Real Algebraic Numbers':" Put German title first as recommended.

Thanks! — Bilorv_(c)(talk) 21:06, 6 August 2018 (UTC)

Concerning "Cantor restates this theorem...": Moved existing reference; added missing reference. —RJGray (talk) 21:44, 6 August 2018 (UTC)

Excellent. — Bilorv_(c)(talk) 22:56, 6 August 2018 (UTC)

Concerning "Cantor's article is short": Added ref to justify use of the word "short". Changed "not even" to "less than"—I feel that a positive phrase here is better than one with a negative in it. I think that having the "short" is important. Readers are more likely to remember "short" than "less than four and a half pages". Also, the fact that Cantor's article is short is one of its characteristics and arises from Weierstrass' desire that Cantor should quickly publish the theorem on the countability of the real algebraic numbers. —RJGray (talk) 18:58, 7 August 2018 (UTC)

Concerning "The proof of Cantor's second theorem does not state why some limits exist. The proof he was using does.": Please look this over again. The section it is in has the structure:

• Bullets about facts that historians have discovered about Cantor's article.
• Then the "lead in": "To explain these facts, historians have pointed to the influence of Cantor's former professors, Karl Weierstrass and Leopold Kronecker."
• This is followed by explanations of the facts, which goes into detail about the influence of Weierstrass and Kronecker on the bulleted facts.

The explanation of the fact that you are concerned about is in the following paragraph:

Kronecker's influence appears in the proof of Cantor's second theorem. Cantor used Dedekind's version of the proof except he left out why the limits a_∞ = lim_n → ∞ a_n and b_∞ = lim_n → ∞ b_n exist. Dedekind had used his "principle of continuity" to prove they exist. This principle (which is equivalent to the least upper bound property of the real numbers) comes from Dedekind's construction of the real numbers, a construction Kronecker did not accept.

If my structuring or explanation is not clear to you, please give me suggestions for modifying them. Thanks, — RJGray (talk) 19:51, 7 August 2018 (UTC)

Right, yes. I was confused as I can't access the source given, but that doesn't matter. The structure and explanation is fine, but the original bullet point still seems unnecessarily obfuscated. Some example improvements:

"The proof of Cantor's second theorem does not state why the limits a_∞ and b_∞ exist. The proof it was based on does."

"The proof of Cantor's second theorem was taken from Dedekind, but omitted the "principle of continuity" which proves that a_∞ and b_∞ exist."

"The proof of Cantor's second theorem is based on Dedekind's proof, but omits explanation of why a_∞ and b_∞ exist."

The point is that the reader shouldn't have to look further down to make sense of the bullet point; the explanatory paragraph should just provide extra detail. — Bilorv_(c)(talk) 01:38, 8 August 2018 (UTC)

Your point is excellent. Thanks for your suggestions; I have used a rewording of one of them. I made sure it followed the 2-sentence format of the other bullets. —RJGray (talk) 20:17, 8 August 2018 (UTC)

• Response to User:Bilorv’s comment about the Skolem paradox (I put Bilorv’s words in green):

“In 1922, Thoralf Skolem proved that if the axioms of set theory are consistent" — Which axioms of set theory exactly? Is this referring to a specific collection of axioms (e.g. ZFC), or saying generally "given any set of consistent axioms ..."? (In the latter case, the definite article "the axioms of set theory" is misleading.)

In my opinion, the current article does not need to give a full explanation of the Skolem paradox, so long as the reader can easily follow the links to a longer discussion. An answer to Bilorv's specific question can be seen in “Skolem’s paradox up close and personal”, by Vaughan Pratt. (The Skolem paradox follows from the Zermelo axioms alone and doesn't require the full ZFC set of axioms). The modern statement of the Löwenheim–Skolem theorem as presented in our article speaks of 'any countable first-order theory' so the paradox holds even for a variety of axiomatizations. It appears that the first-orderness is what causes the paradox. The lead of Löwenheim–Skolem theorem says "In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second-order logic." But these refinements come from later. The purpose of mentioning Skolem's paradox in the current article is surely to show what happened to the notion of 'countability' as employed by Cantor in the work of later mathematicians. EdJohnston (talk) 02:26, 8 August 2018 (UTC)

Right, but I wasn't suggesting that a full explanation should be given. The problem was the ambiguity. I'm happy if "the axioms" is replaced with "standard axioms" or "Zermelo axioms" or "ZFC axioms" or "certain axioms (e.g. ZFC)" etc. But the paragraph needs to be meaningful and fully accurate when read in isolation, so I still think something needs to be changed. — Bilorv_(c)(talk) 15:39, 8 August 2018 (UTC)

Oops, I forgot that "the axioms" was changed to "conventional axioms" a while ago. That addresses my concern. — Bilorv_(c)(talk) 21:42, 8 August 2018 (UTC)

Concerning "Is the 1870 Heine article relevant to Cantor or his work": Put in that Heine was one of Cantor's colleagues. Being a colleague of someone who has had problems with Kronecker would have made Cantor cautious. As for Kronecker's exact reasons for the delay of Heine's article, I know of no records from that time that document the reasons. However, later Harold M. Edwards who has studied Kronecker's work thought it could be due to Heine's study of arbitrary trigonometric series. Instead of speculating on Kronecker's motives for Heine's article delay, I prepare the reader at the end of the last section for Kronecker by stating he had strict views on what is acceptable in mathematics and by going into more detail in a footnote. This should give the reader some idea of Kronecker's thinking. —RJGray (talk) 21:23, 8 August 2018 (UTC)

Yes, this makes sense. — Bilorv_(c)(talk) 21:42, 8 August 2018 (UTC)

Concerning "Liouville's theorem that there are transcendental numbers": I wrote both parts almost identical to the way it appears in Cantor's article and letter. If I understand you correctly, you would like the part in the article to read: "Cantor observes that combining his two theorems yields a new proof of Liouville's theorem that every interval [a, b] contains infinitely many transcendental numbers." The part in the letter would be unchanged: "It would be nice if it could be answered; for example, provided that it could be answered no, one would have a new proof of Liouville's theorem that there are transcendental numbers." The only problem here is that Liouville's theorem is expressed two different ways: one asserting infinitely many transcendental numbers and the other asserting the existence of transcendental numbers. The first implies the second, and the second implies the first as soon as one realizes that the existence of one transcendental implies the existence of infinitely many (which goes back to my earlier mention of generating infinity many by adding rationals to the transcendental that was proved to exist). We are stuck with these choices because Cantor knows that if you proved one statement, the other statement is an easy consequence, so he feels comfortable informally using the two differing statements as if they were equivalent. Which option do you prefer (leaving it unchanged or using Liouville's theorem in both places) or do you have another option? —RJGray (talk) 15:05, 9 August 2018 (UTC)

Yes, I'd rather have Liouville's theorem used to describe both (and I think the linking you just used is better than the article's "Liouville's theorem that there are transcendental numbers", which is too long a phrase to link). — Bilorv_(c)(talk) 15:28, 9 August 2018 (UTC)

Concerning "The proofs": My approach was to stay within the guidelines of WP:Scientific citation guidelines#Examples, derivations and restatements whose first paragraph states:

"Wikipedia is neither a textbook nor a journal. Nonetheless, in mathematics and the mathematical sciences, it is frequently helpful to quote theorems, include simple derivations, and provide illustrative examples. For reasons of notation, clarity, consistency, or simplicity it is often necessary to state things in a slightly different way than they are stated in the references, to provide a different derivation, or to provide an example. This is standard practice in journals, and does not make any claim of novelty.[1] In Wikipedia articles this does not constitute original research and is perfectly permissible – in fact, encouraged – provided that a reader who reads and understands the references can easily see how the material in the Wikipedia article can be inferred. Furthermore, copying extensively from a source with only minor modifications is not normally permitted by copyright law, unless the source has a free license."

In particular, I wrote "The proofs" knowing that providing a different derivation or an example is "perfectly permissible" and is "in fact, encouraged" as long as a reader knowledgeable about the references (in this case, Cantor's 1874 article) can easily see how the Wikipedia derivation can be inferred.

The table in the first proof is an "illustrative example". This example has the additional feature of being able to be verified by doing calculations to generate the polynomials with height ≤ 4, identify the irreducible polynomials, and compute the roots.

The closed interval simplification in the second proof is a slightly "different derivation" that "a reader who reads and understands the references can easily see how the material in the Wikipedia article can be inferred." In fact, I help the reader by describing the major difference between Cantor's derivation and the one using the closed interval simplification.

My reason for the closed interval simplification is for "clarity" and "simplicity" so the Wikipedia proof is more accessible to Wikipedia readers than Cantor's original proof, which was meant for research mathematicians. For example, the closed interval simplification is crucial to the case diagrams, which handle some criticism from the GA review of an older version of this Wikipedia article. Cantor's method of bouncing between closed intervals and their interiors would lead very cluttered diagrams. By the way, it was natural for Cantor to only work with closed intervals—he only had notation for them. He later defined "closed set". Open sets came much later—they were first defined in print by Baire in 1899. (See Gregory Moore's The emergence of open sets, closed sets, and limit points in analysis and topology.)

The example of Cantor's construction is another "illustrative example" with the additional feature of being able to be verified by doing some calculations. — RJGray (talk) 16:12, 10 August 2018 (UTC)

I retract the comment about the table and the closed intervals.

Can you describe to me how the Example of Cantor's construction was written? If you mean to say that you claim the entire section falls under WP:SCG, then I'll have to take a closer look and research into what the standard is in this area. I wouldn't describe it as a "simple derivation" or a "routine calculation", not due to complexity but due to the length of the section. The table is trivial, but not the proof of x_i ∉ (a_n, b_n). But then, surely there's a source which covers this step that Cantor omitted. — Bilorv_(c)(talk) 18:58, 10 August 2018 (UTC)

I came up with a short proof that that for all n, (a_n, b_n) excludes x₁, … , x_2n. This implies Cantor's statement x_n does not belong to the interval (a_n, b_n). Your comment on looking for a source helped me even though I couldn't find a source that had a proof. Discussions of Cantor's 1874 method seem to be fairly rare; nearly everyone seems to prefer discussing the diagonal method. However, seeing again that Cantor made the unproven observation x_n ∉ (a_n, b_n) led me to think that he probably left out the proof because it was a simple proof. In his book, Dauben approaches the result differently. He points out, but doesn't prove, that if a_∞ = x_k, then for sufficiently large index n, x_k would be excluded from all intervals nested within (a_n, b_n). I believe that the new proof fits within the WP:SCG guidelines (it's a simple derivation that makes Cantor's proof more accessible to Wikipedia readers than Cantor's original proof, which was meant for research mathematicians). Also, the trimmed down Example section is now strictly an example.

Your observations have led to this very significant improvement of the article. This together with your other suggestions prove that you've done an excellent job as a Good Article reviewer. In the next few days, I plan to finish working through your remaining suggestions. —RJGray (talk) 23:14, 14 August 2018 (UTC)

I'm very happy with the new version, and glad that you appreciate my comments. The article is in pretty good shape! Just a couple of small things left. — Bilorv_(c)(talk) 00:06, 15 August 2018 (UTC)

Added asymptotic complexity of computer programs computing digits of a transcendental to a footnote. Changed wording of sentence on article's legacy in Lead. Kept it general and changed "namely" to "which includes" which is more accurate. I didn't follow suggestion to mention Cantor's later work because the legacy section mentions quite a few mathematicians. I have mixed feelings about "significant impact" vs. "impact". Either way is fine with me; I just used the more conservative "impact".

It looks like all suggested improvements have been made. The first "Lead" item: The link to "Georg Cantor" has already been done, but wasn't crossed out. —RJGray (talk) 18:53, 17 August 2018 (UTC)

I've changed the complexity reference to a note (in this edit), but feel free to revert me or make other changes if you wish. That is indeed everything, so thanks for all your hard work, I hope you think the article is at a higher standard than it was before the review and that's a pass for GA! — Bilorv_(c)(talk) 22:20, 17 August 2018 (UTC)

I agree that the complexity reference should be a note, thanks for catching that. I especially wish to thank you for all your hard work on the review. It's been a pleasure working with you and I do think the article is at a higher standard than it was before the review. —RJGray (talk) 00:23, 18 August 2018 (UTC)