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The following is an archived discussion of the DYK nomination of the article below. Please do not modify this page. Subsequent comments should be made on the appropriate discussion page (such as this nomination's talk page, the article's talk page or Wikipedia talk:Did you know), unless there is consensus to re-open the discussion at this page. No further edits should be made to this page.

The result was: promoted by Cwmhiraeth (talk) 06:23, 3 December 2018 (UTC)

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Georg Cantor's first set theory article

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Georg Cantor, c. 1870.
Georg Cantor, c. 1870.
  • ... that some mathematicians have disagreed about whether Georg Cantor's proof of the existence of transcendental numbers is constructive or merely existential?

Proof is constructive: Dasgupta 2014, p. 107; Sheppard 2014, pp. 131–132. Proof is non-constructive: Jarvis 2014, p. 18; Chowdhary 2015, p. 19; Stewart 2015, p. 285; Stewart & Tall 2015, p. 333.

Created/expanded by RJGray (talk) and Michael Hardy (talk). Nominated by Michael Hardy (talk) at 04:21, 19 August 2018 (UTC).

  • Not a review, but you should link and boldface the article in the hook. Catrìona (talk) 05:03, 19 August 2018 (UTC)
  • Recently promoted GA article, so clearly meets the article criteria. Original hook is rather technical; I provided a substitute. The QPQ check didn't turn up a review by the nominator - please provide a link to a review. RockMagnetist(talk) 15:31, 24 August 2018 (UTC)
@RockMagnetist: I did this review. Michael Hardy (talk) 22:10, 24 August 2018 (UTC)
  • On further investigation, it appears that the nominator, despite over 200,000 edits, has never nominated a DYK article before. Welcome to DYK, Michael Hardy! RockMagnetist(talk) 19:27, 24 August 2018 (UTC)
  • I formatted the hook to add "(pictured)". However, there is number confusion between "a transcendental number" and "they" — should the end of the hook be "or merely proves it exists"? Yoninah (talk) 21:36, 5 September 2018 (UTC)
I'm not the submitter of this DYK, but in my opinion the meaning would be conveyed by changing the last two words of the hook. You could change "..they exist" to "..such numbers exist". EdJohnston (talk) 21:48, 5 September 2018 (UTC)
@EdJohnston: thanks, I like that. Waiting for the nominator to weigh in... Yoninah (talk) 21:53, 5 September 2018 (UTC)
@Yoninah: @EdJohnston: The singular in "how to construct a transcendental number" is appropriate since in this context one would construct them one at a time, and the plural is appropriate in "they exist" since the proof shows that many transcendental numbers exist, not just one. Whether it says "they exist" or "such numbers exist" doesn't seem immensely important to me, since the meaning of "they" seems clear from the context. I suppose erring on the side of caution one should be explicit and say "such numbers exist." Michael Hardy (talk) 23:14, 5 September 2018 (UTC)
  • Thank you. Restoring tick per RockMagnetist's review. Wait a minute — the review didn't confirm that the five main DYK criteria have been met. Here is a full review: GA approved. New enough, long enough, well referenced, neutrally written, no close paraphrasing seen in online sources. Hook ref verified and cited inline. Images in article and hook are freely licensed. No QPQ needed for nominator with less than 5 DYK credits.
  • Tweaked hook wording:
  • ALT1a: ... that mathematicians do not agree on whether a proof in Georg Cantor's (pictured) first set theory article actually shows how to construct a transcendental number, or merely proves that such numbers exist?
  • ALT1a good to go. Yoninah (talk) 00:05, 6 September 2018 (UTC)
  • Pulling from prep, with due apologies. @RJGray, Michael Hardy, Yoninah, and Cwmhiraeth: The hook might be okay, but the article is not sufficiently referenced. Indeed, I'm a little confused about how it passed the GA review, because though it's well written and I have no reasons to doubt its accuracy, the second half of "The development of Cantor's ideas" is completely unreferenced. Moreover, this raises neutrality issues, because the article seems to take the view that the proof is a constructive proof ("Some mathematicians have attempted to correct this misunderstanding of Cantor's work.") but this statement depends on the previous section, which, as I said, is not adequately referenced. Vanamonde (talk) 02:36, 13 September 2018 (UTC)
  • This was stuck in limbo as it was not re-listed. Needs a new reviewer to address concern(s) raised. Alex Shih (talk) 17:25, 14 September 2018 (UTC)
    @Alex Shih: Apologies for not relisting it, but really this doesn't need a reviewer, it needs work from the creator/nominator. If the sourcing issues were fixed I'd put this back into prep myself. Vanamonde (talk) 18:45, 14 September 2018 (UTC)
@Alex Shih: By the "creator" do you mean those who created the page, as opposed to the DYK hook? By "nominator", do you mean the person who nominated it for DYK? (That's me.) By "creator/nominator", do you mean the creator _or_ the nominator? Or do you mean it needs both? An earlier posting to my talk page left me with an impression that the issues had already been addressed. Michael Hardy (talk) 20:55, 14 September 2018 (UTC)
@Michael Hardy: I think you meant to ping me, rather than Alex. I meant that someone needs to address the sourcing issues I have raised immediately above Alex Shih's comment. The article creator and you are the people best placed to do this, and the people who have an interest in fixing the issue so that the DYK nomination may move forward. This is entirely separate from the message Yoninah left you nine days ago; these are issues I raised two days previously. Vanamonde (talk) 21:17, 14 September 2018 (UTC)
@Vanamonde93: ok, I think that point is becoming clear. Just to be clear about something else: I am the original article creator, although by now RJGray is to a large extent also the creator. He knows more about the sources and history than I do. Michael Hardy (talk) 22:09, 14 September 2018 (UTC)
@Vanamonde93: Concerning the sourcing issue about which you state: "Indeed, I'm a little confused about how it passed the GA review, because though it's well written and I have no reasons to doubt its accuracy, the second half of "The development of Cantor's ideas" is completely unreferenced." In the GA Review, the issue of references for derivations and examples did come up with respect to the section "The Proofs". This issue is the same as the one you are raising for the second half of "The development of Cantor's ideas" (in this half of the section, I am comparing the derivations in two different proofs). Here is how I handled the issue during the GA Review (next 2 paragraphs I wrote for the GA Review):
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."
After receiving my response, the GA reviewer dropped this issue from his list of issues. --RJGray (talk) 00:43, 15 September 2018 (UTC)
I don't know about that, RJGray. It's fine to simplify a proof, but even so you should cite the original. At this point, a reader who wishes to know where that proof came from has no way to find it, which is the basic purpose of WP:V. I'm not saying you should cite every sentence, but you should cite the proof. You can add explanatory footnotes with the source if required. Vanamonde (talk) 00:58, 15 September 2018 (UTC)

What is the current status of this thing? Is it in a queue to appear under "Did you know . . . ?" on the main page? Michael Hardy (talk) 22:31, 16 September 2018 (UTC)

@Michael Hardy: No, it is not. As I said above, the article needs to have references for all the proofs, to comply with our policy on verifiability. Once the required references have been added, I will place this in a queue. Vanamonde (talk) 01:26, 17 September 2018 (UTC)
@Vanamonde93:I have supplied the references for the proofs. Because Cantor was writing a research article for researchers, he left out simple proofs. To handle this, I did the same as I had done earlier on the proof of his uncountability theorem. I point out that he did not supply a proof and then provided a proof for Wikipedia readers. Of course, I do supply a reference to where he states the result that he does not bother proving. --RJGray (talk) 16:44, 17 September 2018 (UTC)
@RJGray: Okay, I've looked at your changes. They are an improvement, but they do not entirely address my concern. My point is fairly simple; Wikipedia is not a scholarly publication. We collate information; we do not provide new information. As such, if we are presenting a proof, it must be based on a proof published in a reliable source elsewhere. We cannot publish our own proofs, even if the editor writing said proof is quite capable of doing so without error. It is possible that Wikiproject Mathematics sees this differently, in which case I'd like to see a link to that, but I cannot see how WP:V can be satisfied any other way. Please don't take this personally; this sort of mixup often occurs when content written and reviewed by editors familiar with a specific topic is brought to wider scrutiny. If you disagree with this, please feel free to request further feedback at WT:DYK. Also, since Michael Hardy has been blocked indefinitely, I'm afraid this falls entirely on your shoulders at the moment. Regards, Vanamonde (talk) 19:32, 17 September 2018 (UTC)
@Vanamonde93: Thank you for your feedback. Don't worry about me taking it personally. I like to get things right and I don't mind differences of opinion—I've handled them before. In fact, one of the pleasurable things about Wikipedia is that differences of opinion are handled with respect. Unfortunately, there are too many websites where this isn't true.
I hope you are not in a rush to settle this issue. Thanks for mentioning the Wikiproject Mathematics site. I plan to bring up the issue there after researching it a bit. Unfortunately, I didn't have much time last week to devote to studying the relevant Wikipedia policies.
This issue is important for me to because I suspect it will occur in the future. I tend to work in history of mathematics and may be faced again with the fact that research mathematicians tend to leave out simple derivations or proofs when communicating with their fellow research mathematicians via articles and letters. The only reason that I provided my own simple derivations was because Cantor left out two: one in his article and the other in a letter to Dedekind. I could follow Cantor and skip the derivations. However, because Wikipedia appeals to a wide audience, I wanted to make sure readers had a complete proof rather than expecting them to finish it. --RJGray (talk) 17:18, 23 September 2018 (UTC)
@RJGray: I am not in a hurry, though we should try to keep it to a reasonable schedule to be fair to the DYK process. Thanks for not taking it personally. Your position about wanting readers to have a complete proof is quite reasonable, and I'm certainly not suggesting that your derivation is wrong, only that it wouldn't be acceptable on its own even if it were entirely correct, without backing from a reliable source. Regards, Vanamonde (talk) 17:23, 23 September 2018 (UTC)
@Vanamonde93: If I compute 5283 × 6117 and get 32,316,111, it may be that no "reliable source" can be cited for the value of the product of those two numbers. But the technique is taught in elementary school. Something similar applies in many cases to routine algebraic derivations in Wikipedia articles. The is then the question of where to draw the line between that sort of thing and original research. Michael Hardy (talk) 19:19, 23 September 2018 (UTC)
@Michael Hardy: I would not ask for a source for routine arithmetic, per WP:CALC. However, I would draw the line for requiring reliable sources somewhere between a single-step computation that anyone with access to the internet is capable of performing on their device, and Georg Cantor's theorems. I firmly believe those require a source, and that presenting an editor's own derivation isn't enough. If you disagree, I suggest you invite comments from other editors at WT:DYK. Vanamonde (talk) 19:38, 23 September 2018 (UTC)
@Vanamonde93: Thank you for your patience. Even though I haven't changed my opinion that sources are not needed in this case, I decided that finding sources would be a good challenge of my knowledge of the literature. It took a bit of work but I suceeded. I have updated the Wikipedia article.
The source I found for my proof of the existence of transcendental numbers is Perron's book, which is in German and has not been translated. However, in my research on Wikipedia policies, I learned how to handle this: I've put the German proof and my translation of it in a note (see WP:V#Quoting). The source I found for my proof of Cantor's uncountability theorem is the article "Georg Cantor and Transcendental Numbers".
I must say that I am happier having the sources than not having them—I like to saturate Wikipedia articles with citations. I regard citations as doorways to deeper knowledge of a subject.
However, I think that Wikipedia's verifiability policy of citations and reliable sources can be characterized as "passive verifiability". By this I mean that the citations mostly sit there passively with few people actively looking many of them up. Consider my article with its 63 references with one reference containing 6 citations. How many citations will the typical reader of an article look up? Probably far fewer than 63. On the positive side, the citations are available if issues come up.
Now consider an article's mathematical proofs. Proofs actively engage readers who think through them and decide if they are correct or not. And if they aren't correct, they can fix them. So proofs are an example of "active verifiability". I always read mathematical proofs carefully and when I find one that is inaccurate or not clear, I rewrite it. So the accuracy of a mathematical proof depends not just on the person who initially writes it. Its verifiability is increased by everyone who reads it, thinks about it, and improves it.
On the other hand, I seldom look up the citations of a Wikipedia article unless I'm interested in reading further on the subject. I know it's hard work checking sources—the last thing I do before posting an article I wrote is to check all my citations for accuracy.
Another weakness of Wikipedia's verifiability policy is that reliable sources may have errors. The section "The disagreement about Cantor's existence proof" gives examples. In fact, the books asserting that Cantor's existence proof is non-constructive outnumber the books asserting asserting his proof is constructive.
I wish to thank you for suggesting your improvement. This article has now benefited from two GA reviews (it failed the first one but that motivated me to do a second rewrite of the article) and one DYK review. —RJGray (talk) 22:59, 26 September 2018 (UTC)
@RJGray: Thanks for the sources. Unless I'm missing something, though, the "first theorem" still doesn't have citations. Also: I'm understanding, from what you said above, that you're using the same source for all of the second theorem (footnote A). If this is the case, please duplicate it (you can just copy the footnote) at the end of all of the relevant paragraphs; else this is going to run into trouble again when it's in prep. In general, even if you're using the same source for multiple paragraphs, duplicate it at the end of each paragraph. As to your other points, I agree in theory, but the fact is that even though I have more mathematical education than probably 99% of the world's population, I would miss most errors in a proof such as this one; which is why we need the refs, even if they aren't followed up on. Vanamonde (talk) 01:14, 27 September 2018 (UTC)
(GA reviewer here and I've been lurking for a few days.) Per WP:CITEDENSE, there's no "one citation per paragraph" rule on Wikipedia. Repeating a citation every paragraph is an optional stylistic choice. If a user would remove this from the DYK queue for that reason then they would be in the wrong. As for the "first theorem" section, if my memory is correct then I recall that it's a summary of parts of the Cantor article, vacuously sourced to that article, along with a bit of WP:SCG/WP:CALC. (Per WP:MINREF it didn't need an inline citation... but the fact you claim it does counts as a "challenge" and actually means it now does, somewhat bizarrely.) — Bilorv(c)(talk) 01:42, 27 September 2018 (UTC)
@Bilorv: Well, in my opinion articles should always have a minimum of one citation per paragraph, but in any case this is actually codified as a DYK rule (supplementary rule D2, if you're interested). In any case, if parts are based on the George Cantor article, then that needs to be fixed. Wikipedia isn't a reliable source, and if that article is in such good shape, then it should be easy enough to copy references over. Vanamonde (talk) 01:55, 27 September 2018 (UTC)
To clarify, when I said "vacuously sourced to that article", I meant "vacuously sourced to the primary source of Cantor's paper", not some other Wikipedia article. — Bilorv(c)(talk) 10:45, 27 September 2018 (UTC)
That's even easier, then: just add that as a source, with an explanatory footnote if necessary, at the end of every paragraphs that's taken from it. Vanamonde (talk) 17:44, 27 September 2018 (UTC)
@Vanamonde93 and Bilorv: I don't want to bother with the question of whether or not I need references in the subsections "First theorem" and "Second theorem". I was surprised that I didn't put them in! I like to encourage my readers to read Cantor's original article. One way I do this is to supply references to his article and to a translation of his article whether they are needed or not. So I added them.
Concerning the DYK supplementary rule D2. This rule states: "The article in general should use inline, cited sources. A rule of thumb is one inline citation per paragraph, excluding the lead, plot summaries, and paragraphs which summarize other cited content." According to Rule of thumb: "The English phrase rule of thumb refers to a principle with broad application that is not intended to be strictly accurate or reliable for every situation." If it was a rigid rule that required one citation per paragraph, I've already came up with two problems such a rigid requirement can create:
(1) Let's say I had an article that I planned to nominate for DYK and one paragraph of my article had no citation but the following paragraph had more than one. Then I might change my paragraphing so the first paragraph would have one citation. Of course, this could negatively affect the writing quality.
(2) Consider the Cantor article. The first paragraph of Georg Cantor's first set theory article#Example of Cantor's construction is a description of a simple example that I made up to illustrate how Cantor's construction works. I am permitted to have such an example by WP:Scientific citation guidelines#Examples, derivations and restatements: "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. … In Wikipedia articles this does not constitute original research and is perfectly permissible …" Since I'm providing the example it has no reference. So if rule D2 was rigid, I'd have to decide which was most important: A "Did you know" or an example that helped my readers.
By the way, I have 67 references in 49 paragraphs, which is 1.37 refs/paragraph. So my article is averaging 37% more refs per paragraph than an article that only contains one ref per paragraph. ≈—RJGray (talk) 01:42, 28 September 2018 (UTC)
@RJGray: I'm a bit busy at the moment so please forgive the short response. First off, right now I'm only seeing a citation for theorem two. Second, the problems with writing style and structure are easily addressed by duplicating references. I feel like I am somehow not getting this point across, but it's very simple, and it addresses your concern entirely. If multiple paragraphs are sourced to page 2 of Cantor's paper, then you just cite page 2 of Cantor's paper after each of those paragraphs. Why is this proving so difficult? Vanamonde (talk) 01:51, 28 September 2018 (UTC)
@Vanamonde93: Sorry, I'm a bit tired and it appears that I didn't submit my changes, which are done now. On paragraphing, there may be a problem on "Second theorem". I see three options on the structure of the paragraph starting: "Either the number of intervals generated is finite or infinite" Option 1: the ending of case 3 is the end of the paragraph (this is where I put a ref). Option 2: each case is a paragraph, so in this option, I would put a ref after every case. Option 3: the sentence following the cases is the end of the paragraph, so in this option, I would put the ref at the end of this sentence. I chose Option 1 because I think the colon implies that the three cases belong to the same sentence as the colon. Which option do you think is best? Take your time answering, I'm getting off the computer now. — Preceding unsigned comment added by RJGray (talk • contribs) 02:37, 28 September 2018 (UTC)
@RJGray: My apologies, I've been inactive for the past few days. Your latest changes look good. My final citation quibble is about the proof provided in "dense sequences", which also appears not to have a source. Since the article does not seem to hinge on this, you could simply temporarily remove the section if you want to move forward with this quickly. Vanamonde (talk) 17:01, 1 October 2018 (UTC)
@Vanamonde93: I'm happy that you are pleased with my latest changes. I've made one further change. In my last response, I mentioned the "Example of Cantor's construction" as not needing a citation for an example. I suspect that quite a few examples in math articles won't have a citation available, but it turns out that this particular example does have one. When I was writing it, I must have been concentrating on the math and forgot that nearly the same example appeared in the literature as an exercise. But returning to it now, I remembered that it looked familiar and remembered the probable source. I've added a note about this.
Concerning dense sequences, I have a suggestion. But first the reasons I would like to keep it:
1. In the next section, I use denseness to prove that Cantor's construction generates infinitely many intervals.
2. It identifies what is mathematically special about the second case.
3. It deepens a reader's appreciation of the three cases because they are regarding them from a different angle.
4. It helped me in my reply to Ipsic's question (see next section of Talk).
5. See last sentence in next paragraph.
My suggestion is to bring in Cantor's second proof of the uncountability of the reals, which he published in 1879. This proof is not generally known, at least not in the US. As far as I know, it hasn't been translated into English, but there is a French translation from the 1880s. His second proof only uses 2 cases: The given set of reals is dense or not dense in the interval [a, b]. In the second case, there's an open interval (c, d) containing no reals from the given set, so any real in this open interval does not belong to the given set (this handles cases 1 and 3 in one step). In ths dense case, he gets a single real number. I think that this addition would add to the comprehensiveness of the article and it's a simple example of a great mathematician viewing his theorem from another angle and developing a new proof.
Of course, it will take me a bit of time to add the new material in a succinct way that meshes nicely with what's there. I hope to keep it down to a short paragraph. The big unknown is his dense case. I didn't have the time today to look at it (I'd like to figure it out on my own before looking at his proof). Also, I will be on vacation from next Wednesday to the following Tuesday and won't be around computers as much, but I will have hours of travel to work on the new material.
Thanks for all your feedback. I do appreciate the work you are doing for this Cantor article and the work you do for Wikipedia. —RJGray (talk) 01:43, 2 October 2018 (UTC)
@RJGray: If you have a citation available, that's great, and you should add it. About denseness: I recognize that there's good reasons for wanting it in the article, but unless Cantor's 1879 paper is implying it very directly, I don't think that's a good route to go down. We're coming back to a fairly basic problem here. I probably have had more education in math than most Wikipedia readers: yet, I could not tell if there was an error in the mathematical argument you put forth above. Thus in the absence of a reference saying that, it isn't really verifiable. Please understand, I'm not blaming you for this; it's quite possibly common practice in the mathematics Wikiproject. But to an outsider, this is a problem. Vanamonde (talk) 01:52, 2 October 2018 (UTC)
I'm afraid this doesn't accord with the text of WP:SCG. It reads "Some statements are uncontroversial and widely known among people familiar with a discipline." It then gives three statements which I do not understand at all, and yet they don't need citations because they are uncontroversial among people familiar with the subject. In the case of dense sequences, I can understand the argument easily using only first year undergraduate knowledge (because that's my current level of education), so it's hardly controversial or requiring of esoteric methods. — Bilorv(c)(talk) 11:16, 2 October 2018 (UTC)
@Bilorv: I'm aware of the guideline, but I'm afraid you're misreading it. The guideline allows some flexibility in where citations need to be placed, and how general they might be. It does not permit completely unsourced content. Some source in the article needs to support the information in it. Vanamonde (talk) 14:50, 2 October 2018 (UTC)
@Vanamonde93: My current plan to eliminate the "Dense sequences" subsection and add a new section "Cantor's second uncountability proof" following the "Example of Cantor's construction" subsection. I figured out Cantor's 1879 proof and now I've starting reading it. He calls it a simplification of his 1874 proof and it is. It only has 2 cases, the old Case 1 and Case 3 are handled at the same time, and Case 2 is the dense sequence case. This happens because he points out that there are two kinds of sets: those that are dense in the interval [a, b] and those that are not. This splits the new proof into two cases at the start. His original proof starts with his construction and the construction divides the proof into three cases. Since I'll be gone for a week, it will take me a minimum of a week and a half, but probably a bit longer since I want the writing to be as good as the rest of the article and this always takes time. So you will have your references and I'll have that Case 2 is the dense set (or sequence) case. Also, readers will learn of Cantor's second proof, which I didn't know about for years. Overall, I think it will improve the article. —RJGray (talk) 19:32, 2 October 2018 (UTC)
@RJGray: Sounds good, ping me when you are done. Vanamonde (talk) 20:14, 2 October 2018 (UTC)
@Narutolovehinata5 and Vanamonde93: My original time estimate was way off. I have finished the translation of Cantor's second uncountability proof of 1879 and have had it corrected by another editor. I am providing interested readers a with wikitable that consists of the English translation of the proof and the German text so they can be read and compared in parallel. However, there seems to be a problem with having a collapsed table in a Notes section, so it appears that I might have to choose between an uncollapsed (large) table or have an ending subsection titled: "Appendix: Cantor's second uncountability theorem with translation". I need to know the Wikipedia guidelines on this. I am now writing a new section "Cantor's second uncountability proof" that will have to be cleanly integrated with a couple of the existing sections. Good news for Vanamonde93: I can use Cantor's 1879 proof as a reference for the proof I gave in Case 2 of the subsection "Second Theorem". My proof appears as part of his 1897 proof, which is no surprise since it's the obvious proof. —RJGray (talk) 23:59, 28 October 2018 (UTC)
@RJGray: The niceties of MOS are not my speciality, but as long as it complies with MOS:TABLE, I think both options could work. I personally have not seen an "appendix" before, so I have a slight preference for the other structure. Vanamonde (talk) 14:06, 29 October 2018 (UTC)
@Narutolovehinata5 and Vanamonde93: I apologize for not keeping you up-to-date. I'm nearly done--mostly polishing and a few other relatively small things to do. I should be done by Sunday or Monday. —RJGray (talk) 18:47, 9 November 2018 (UTC)
@Narutolovehinata5 and Vanamonde93: Thank you for your patience. The big changes are the addition of the new section "Cantor's second uncountability proof" and the removal of subsection "Dense sequences". If I had been familiar with Cantor's 1879 article earlier, I would have made it a part of the Wikipedia article earlier because it's a case of Cantor improving his old proof using the topological notion of everywhere dense in an interval. So it sheds light on the key proof of his 1874 article. I wish to thank Vanamonde93 for his DYK work on this article. It was his comments that led me to the changes I have made. Also, wish to thank Jochen Burghardt for his help in getting a collapsed wikitable in a ref note and for his correcting my errors in my translation of Cantor's 1879 proof. — Preceding unsigned comment added by RJGray (talk • contribs) 23:24, 12 November 2018 (UTC)
Repinging Vanamonde and Narutolovehinata5, since the original ping wouldn't have gone through—pings only work if a sig is included in the edit, and the sig was omitted by RJGray in the above post. BlueMoonset (talk) 15:21, 24 November 2018 (UTC)
@RJGray: Thanks for the update. The provenance of the last two paragraphs of "Cantor's 1879 proof" are still unclear; it is possible you need to copy over a citation from the notes section. Vanamonde (talk) 15:58, 24 November 2018 (UTC)
@Narutolovehinata5 and Vanamonde93: Added citations for the last 2 paragraphs. The first paragraph comes from Cantor's proof. The second paragraph is a description of what happens when Cantor's proof is applied to the Example. So both paragraphs should reference Cantor's proof. —RJGray (talk) 18:04, 26 November 2018 (UTC)
ALT1a is GTG, per previous review and comments above; AGF on offline sources. Vanamonde (talk) 18:25, 26 November 2018 (UTC)
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