Talk:Cantor's first set theory article/GA2: Difference between revisions

Revision as of 15:05, 9 August 2018

Latest revision as of 05:55, 17 December 2019

(11 intermediate revisions by 3 users not shown)

Line 11:

#It is '''reasonably well written'''.

#:a ''(prose, spelling, and grammar)'': {{GAList/check|y}} b ''([[Wikipedia:Manual of Style|MoS]] for [[Wikipedia:Manual of Style/Lead section|lead]], [[Wikipedia:Manual of Style/Layout|layout]], [[Wikipedia:Manual of Style/Words to watch|word choice]], [[Wikipedia:Manual of Style/Writing about fiction|fiction]], and [[Wikipedia:Manual of Style/Embedded lists|lists]])'': {{GAList/check|y}}

#::

#It is '''factually accurate''' and '''[[Wikipedia:Verifiability|verifiable]]'''.

#:a ''([[Wikipedia:Manual of Style/Layout#Notes and references|reference section]])'': {{GAList/check|y}} b ''(citations to [[Wikipedia:Identifying reliable sources|reliable sources]])'': {{GAList/check|y}} c ''([[Wikipedia:No original research|OR]])'': {{GAList/check|y}} d ''([[Wikipedia:Copyrights|copyvio]] and [[Wikipedia:Plagiarism|plagiarism]])'': {{GAList/check|y}}

#::

#It is '''broad in its coverage'''.

Line 29:

#::

#'''Overall''':

#:''Pass/Fail'': {{GAList/check|y}}

#:: <!-- Template:GAList -->

Line 39:

===The proofs===

*<s>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.</s>

===The development of Cantor's ideas===

*<s>"[[Liouville_number#Liouville_numbers_and_transcendence|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.</s>

===The disagreement about Cantor's existence proof===

*<s>In the third paragraph, the "exact" asymptotics <math>O(2^{\sqrt[3]{n}})</math> and <math>O(n^2 \log^2 n \log \log n)</math> might as well be mentioned.</s>

*<s>"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 [[Monotone convergence theorem|Monotonic Sequence Theorem]] to deduce that <math>\alpha^\infty</math> and <math>\beta^\infty</math> 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?</s>

*<s>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.)</s>

Line 55:

===Lead===

*<s>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 ..."</s>

*<s>"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.</s>

*<s>"Cantor's article also contains a proof of the existence of transcendental numbers" — I would mention their infinitude as well.</s>

*<s>"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."</s>

Line 109:

: 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. —[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 21:23, 8 August 2018 (UTC)

::Yes, this makes sense. <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''_{[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]}</span> 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 number|Liouville's theorem]] that every interval [''a'',&nbsp;''b''] contains infinitely many [[transcendental number]]s." 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 number|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? —[[User:RJGray|RJGray]] ([[User talk: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 number#Liouville numbers and transcendence|Liouville's theorem that there are transcendental numbers]]", which is too long a phrase to link). <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''_{[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]}</span> 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 [https://www.sciencedirect.com/science/article/pii/S0315086008000050 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. — [[User:RJGray|RJGray]] ([[User talk: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 [[Georg Cantor%27s first set theory article#Example of Cantor's construction|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 {{nowrap|''x_i'' ∉ (''a_n'', ''b_n'')}}. But then, surely there's a source which covers this step that Cantor omitted. <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''_{[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]}</span> 18:58, 10 August 2018 (UTC)

:I came up with a short proof that that for all&nbsp;''n'', (''a''_''n'',&nbsp;''b''_''n'') excludes ''x''₁,&nbsp;…&nbsp;,&nbsp;''x''_2''n''. This implies Cantor's statement ''x''_''n'' does not belong to the interval (''a''_''n'',&nbsp;''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&nbsp;&notin;&nbsp;(''a''_n,&nbsp;''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''_∞&nbsp;=&nbsp;''x''_''k'', then for sufficiently large index ''n'', ''x''_k would be excluded from all intervals nested within (''a''_n,&nbsp;''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. —[[User:RJGray|RJGray]] ([[User talk: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. <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''_{[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]}</span> 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. —[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 18:53, 17 August 2018 (UTC)

::I've changed the complexity reference to a note (in [[Special:Diff/855385322|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'''! <span class="nowrap">— '''[[User:Bilorv|Bilorv]]'''_{[[Special:Contribs/Bilorv|(c)]][[User talk:Bilorv|('''talk''')]]}</span> 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. —[[User:RJGray|RJGray]] ([[User talk:RJGray|talk]]) 00:23, 18 August 2018 (UTC)