User:RJGray/Translate

Cantor's first set theory article

Read: User:RJGray/Translate BEFORE WRITING TO IRY-HOR.

User talk:Iry-Hor/Archive 11

This article is about Cantor's first article on infinite sets, which contains his discovery of two kinds of infinite sets: countable sets and uncountable sets. The members of a countable set, such as the fractions between 0 and 1, can be written as a sequence, for example 1/2, 1/3, 2/3, 1/4, 3/4, …. The members of an uncountable set, such as the real numbers between 0 and 1, cannot be written as a sequence. The significant developments in mathematics that came from the use of countable and uncountable sets justify the importance of this article. Also, it would be good to have another featured article on mathematics: of the 5752 featured articles, only 18 (about 0.3%) are on mathematics.

Number of featured articles:
 * Featured articles (4/5/2020: 5,741) (4/19: 5740) (4/24: 5745) (4/28: 5752)
 * English_Wikipedia (4/5/2020: 6,866) [Updated daily by a bot NOT!!]

Cantor's first set theory article Refs in lead

Translate Cantor draft

Featured_articles Cantor's Absolute Infinite Acta Mathematica

Axiom of limitation of size Von Neumann–Bernays–Gödel set theory Sandbox100 Sandbox101 Sandboxcantor1 Sandboxcantor2

Math TODO:

MOS:MATH WikiProject Mathematics

MOS:ACCESS MOS:TABLECAPTION

Georg Cantor Cantor's diagonal argument Controversy over Cantor's theory  Axiom of dependent choice Function (set theory) GA_Review Representation theory of finite groups Lorentz group Axiom schema of replacement set theory ZFC Zermelo set theory

WP:FAC WP:Featured articles WP:Good articles WP:Mentoring for FAC Displaying a formula

File copyright tags/All

Talk:Georg Cantor's first set theory article/GA2 User_talk:D.Lazard User_talk:RJGray

WP:Verifiability, not truth WP:Good articles

https://en.wikiversity.org/wiki/WikiJournal_of_Science



Reason for ref [2] in Lead
BE SURE TO WRITE TO Iry-Hor about it, esp. part about experiment!! FROM MOS:LEADCITE:

"The lead must conform to verifiability, biographies of living persons, and other policies. The verifiability policy advises that material that is challenged or likely to be challenged, and direct quotations, should be supported by an inline citation. ... The necessity for citations in a lead should be determined on a case-by-case basis by editorial consensus. Complex, current, or controversial subjects may require many citations; others, few or none. The presence of citations in the introduction is neither required in every article nor prohibited in any article."

As can be seen from the history of this sentence, the original sentence appeared on February 12, 2016 and was removed on April 2, 2016. I then rewrote the sentence and add a citation for it. On December 16, 2019, I removed 4 citations in the lead, keeping only the citation for a sentence containing a direct quotation. I did this while I was working with Iry-Hor (my mentor) to get this article ready for Featured Article nomination. As an experiment, I removed these 4 citations to see if anyone reacted to the lack of a citation. This happened on April 17, 2020 when the sentence about determining the nature of Cantor's proof was weakened to use "might" instead of "can". Thinking about it, I realized that this weakening was valid since "can" does make strong claim that needs a citation. This was the second time this sentence or a similar sentence has been challenged. So it has been challenged twice and both times a citation was successfully used to handle the challenge. The last time the sentence was unchallenged for 3 years, 8 months, but when I removed the citation as an experiment, it took only 4 months to be challenged. Here's the history:
 * Original sentence added, then removed:
 * A careful study of Cantor's article will determine whether or not his proof is constructive.
 * February 12, 2016‎ RJGray: Created page with 'thumb|Georg Cantor, c. 1870. Georg Cantor's first set theory article ...'
 * April 2, 2016‎ William M. Connolley: rm interpolated sentence which is (a) unreffed and (b) unlikely to be true, if there is a genuine disagreement


 * New sentence added with citation, then removed citation as an experiment:
 * Since Cantor's proof either constructs transcendental numbers or does not, an analysis of his article can determine whether his proof is constructive or non-constructive.[Gray 1994, p. 819–821.]
 * April 12, 2016‎ RJGray: Added a replacement for sentence that was removed. See Talk.
 * December 16, 2019‎ RJGray: Rewrote to conform to WP:MOS/LEAD. Removed all citations [in Lead] except one for direct quote: see WP:MOS/LEAD#Citations.


 * Sentence modified ("can" changed to "might"), then changed back to April 12, 2016‎ sentence:
 * Since Cantor's proof either constructs transcendental numbers or does not, an analysis of his article might determine whether his proof is constructive or non-constructive.
 * April 17, 2020‎ Ore4444: [Left out comment: Changed "can" to "might".]
 * April 23, 2020‎ RJGray: Changed "might" back to "can" and also added a ref. Thanks for pointing out the need for a ref.

DON'T INCLUDE WITH ABOVE! FROM Iry-Hor in User talk:RJGray:
 * As a MOS rule, the lead should have no references, except for statements that are highly contentious. This is not the case here so all refs of the lead should be removed. This would be raised at FAC immediately.
 * Manual of Style/Lead section states that:
 * "The lead must conform to verifiability, biographies of living persons, and other policies. The verifiability policy advises that material that is challenged or likely to be challenged, and direct quotations, should be supported by an inline citation."
 * This covers my use of the direct quotation "Cantor's revolutionary discovery". As for the other inline citations, I've deleted them but I will put back any that get challenges on the statement that it protects.

Michael Hardy
Hi Michael, The article you started years ago (under the name "Cantor's first uncountability proof") will soon be on its way to (hopefully) becoming a Featured Article. Iry-Hor is an excellent mentor. He gave me a list of 16 items to fix or consider. I made a number of changes and he now advises me to nominate the article. I will be waiting until January when I will have the time to give quick responses to the reviewers. The latest copy of the article can be found at User:RJGray/Sandbox100.

One place that needed changing was the lead. The current lead does not include the article title in bold, which the MOS requires. Iry-Hor suggested a couple of changes that put the article title at the beginning. Here's the first sentence of the new lead:


 * Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties.

The article title has been changed to start with "Cantor's" rather than "Georg Cantor's". This has several advantages:
 * WP:Article titles states: "The goal of conciseness is to balance brevity with sufficient information to identify the topic to a person familiar with the general subject area." It goes on to state "Exceptions exist for biographical articles. For example, neither a given name nor a family name is usually omitted or abbreviated for conciseness." The Cantor article doesn't qualify for this exception. Since I'm trying to meet the standards for a featured article, it's best for me to follow the article title rules.
 * It conforms to the article titles used for other mathematicians' works. For example, there are article titles beginning with "Zermelo's", "Dedekind's", and "Von Neumann's". Of these, only "Von Neumann's" has a redirect: "John von Neumann's inequality". Since  these article titles all begin with "Von Neumann's", a search gives all files beginning with "Von Neumann's", whereas a search on "Cantor's" currently leaves out "Cantor's first set theory article"—it's left out since it's a redirect rather than an article title. I use this type of search to delve deeper into a mathematician's work.
 * A featured article should have the best possible opening. A repeat of "Georg Cantor" in the first sentence is redundant. Also, without the repeat, the wikilinked "Georg Cantor" is the first occurrence of "Georg Cantor".

By the way, the current article title starts with "Georg Cantor's" because when I rewrote the article, I made the first mistake in WP:MOS: "Links should not be placed in the boldface reiteration of the title in the opening sentence of a lead". You corrected that mistake. I'm now an avid reader of the MOS since I'm working on making it a featured article.

I thought it might be a good learning experience for me to handle the page move from "Georg Cantor's first uncountability proof" to "Cantor's first uncountability proof". However, I've already learned that I'll have to make a technical request because the "Cantor's first set theory article" redirect page has two items in its history. Could you make the move for me? My reason for the move is: "Article title change. New title is more concise."

Thank you for all the help and encouragement you've given me since I started writing for Wikipedia. In fact, your Cantor article nominations for GA and DYK last year made me interested in nominating it for FA. —~

Iry-Hor
Manual of Style/Mathematics WikiProject Mathematics/Proofs Help:Pictures Cantor's 1874 uncountability proof

Today's Featured Article: Check when a math article last appeared. Go to WP:TFA. RJGray I think your article is really excellent. You can confidently propose it at FAC now! Write a very short blurb to present it at FAC and go for it. I will support the article and I am sure plenty of other people will do so too. As I said, once this is FA (FAC can take a couple of months so it will be next year), I strongly suggest you propose it at Today's Features Article Candidates.Iry-Hor (talk) 08:34, 5 December 2019 (UTC)

Hi Iry-Hor, That was fast! I didn't realize I was so close to a FA. I was going to ask you for an example blurb, but I've found them myself. For example, I found one at the beginning of Featured article candidates/Featured log/November 2019. With the holiday season approaching, I'm going to be very busy so I won't have that much time to fix things. I have no idea of what kind of response time they expect on a FAC. Should I wait or go ahead? Of course, I first have to write the blurb and knowing myself, I will read quite a few blurbs first, and then write and rewrite mine quite a few times. Thanks again for the work you have done in making excellent suggestions for the article. -RJGray (talk)

Ferreiros
The Early Development of Set Theory: The Stanford Encyclopedia of Philosophy (Summer 2019 Edition)

Although Cantor might have found that paradox as early as 1883, immediately after introducing the transfinite ordinals (for arguments in favour of this idea see Purkert & Ilgauds 1987 and Tait 2000), the evidence indicates clearly that it was not until 1896/97 that he found this paradoxical argument and realized its implications. By this time, he was also able to employ Cantor’s Theorem to yield the Cantor paradox, or paradox of the alephs: if there existed a “set of all” cardinal numbers (alephs), Cantor’s Theorem applied to it would give a new aleph ℵ, such that ℵ<ℵ. The great set theorist realized perfectly well that these paradoxes were a fatal blow to the “logical” approaches to sets favoured by Frege and Dedekind. Cantor emphasized that his views were “in diametrical opposition” to Dedekind’s, and in particular to his “naïve assumption that all well-defined collections, or systems, are also ‘consistent systems’ ” (see the letter to Hilbert, Nov. 15, 1899, in Purkert & Ilgauds 1987: 154). (Contrary to what has often been claimed, Cantor’s ambiguous definition of set in his paper of 1895 was intended to be “diametrically opposite” to the logicists’ understanding of sets—often called “naïve” set theory, or can more properly be called the dichotomy conception of sets, following a suggestion of Gödel.)

Cantor thought he could solve the problem of the paradoxes by distinguishing between “consistent multiplicities” or sets, and “inconsistent multiplicities”. But, in the absence of explicit criteria for the distinction, this was simply a verbal answer to the problem. Being aware of deficiencies in his new ideas, Cantor never published a last paper he had been preparing, in which he planned to discuss the paradoxes and the problem of well-ordering (we know quite well the contents of this unpublished paper, as Cantor discussed it in correspondence with Dedekind and Hilbert; see the 1899 letters to Dedekind in Cantor 1932, or Ewald 1996: vol. II). Cantor presented an argument that relied on the “Burali-Forti” paradox of the ordinals, and aimed to prove that every set can be well-ordered. This argument was later rediscovered by the British mathematician P.E.B. Jourdain, but it is open to criticism because it works with “inconsistent multiplicities” (Cantor’s term in the above-mentioned letters).

Michael Hardy: Addition to article & Featured article nomination
Hi Michael, I've decided that using Cantor's 1874 uncountability proof is shorter and better than using Cantor's first published uncountability proof. I've also changed Cantor's second uncountability proof to Cantor's 1879 uncountability proof. I plan to change it at all 16 references. Shall I play it safe and keep the

Hi Michael,

I've added another proof to Georg Cantor's first set theory article. The current Wikipedia article contains two of Cantor's three pre-diagonal uncountability proofs. I've added the third proof, which came before the other proofs but is in a letter that wasn't published until 1937. My changes appear in User:RJGray/Sandbox100. I've added the proof to the section: The development of Cantor's ideas. I also modified the paragraphs that precede and follow the proof.

I also had to change the current lead that states: "This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument." The proof in Cantor's 1874 article is actually Cantor's first published uncountability proof since Cantor's letter contains the first proof but was published later. I now use Cantor's first published uncountability proof. An alternative wording is: Cantor's first proof of uncountability if you think that it's better. I still have to change the references to Cantor's first uncountability proof that occur in 16 other Wikipedia articles.

I revisited the Cantor article because I'm seriously considering nominating it for Featured Article (the new addition makes it more comprehensive). I'm glad you nominated the article for Good Article and DYK. For Featured Article, it's stated that "Nominators must be sufficiently familiar with the subject matter and sources to deal with objections during the featured article candidates (FAC) process.": see WP:FAC. Since I'm most familiar with the sources, I think that I should nominate it.

It's also recommended that I find a mentor before nominating the article. The mentor page (see WP:FAM) lists several editors for math or science:


 * User:Iry-Hor: My Wiki work is exclusively on Ancient Egypt, but I can help in Ancient History, and Mathematics and Physics articles.


 * User:Ceranthor: Science, biography, mythology.


 * User:Casliber: Biology, astronomy, other sciencey stuff, sports, popular culture.


 * User:Cwmhiraeth: I prefer scientific or general topics, and am totally uninterested in popular culture.

I welcome any suggestions you may have on who would be a good mentor, either one of the above or someone else. I found my experiences with the Good Article and DYK nominations to be a lot of work, but it was well worth it. It improved the article and I learned a lot about writing good Wikipedia articles. I look forward to the suggestions I get from you, from a mentor, and from a featured article review.

Thanks, RJGray (talk) 00:54, 15 February 2016 (UTC)

Going for the gold: Better and best article candidates
Wikipedia has two classifications for high-quality articles that have been through an assessment nomination process: Good and Featured. Below are five places where assessments take place, and you may be able to contribute.

Candidates for Good and even Featured classification may be a long way from perfect. You may find the checklist approach to improving articles described in Chapter 18: Better articles: A systematic approach a big help here. As always, when you're looking over listed articles, you can pick and choose. You don't have to comment on articles you're not interested in, or where you don't see obvious opportunities for improvement.


 * Good article nominations (shortcut: WP:GAN). At any given time, you'll probably find several hundred articles undergoing review, nicely organized into topical categories.
 * Good article reassessment (shortcut: WP:GAR). Good articles occasionally go bad, or turn out never to have been that good. This page is where Good article ratings are reassessed. Typically you see only a handful of articles here at any time. Most reviewers probably visit because of a notice on an article talk page.
 * Featured article candidates (shortcut: WP:FAC). Nominees for Wikipedia's highest quality category are on this page—usually 50 to 100 articles at a time. Articles are often up for a month or two while undergoing review, so checking in every 3 or 4 weeks to see what's up is frequent enough.


 * As with Good article nominations, Featured article candidates have almost always been nominated by the editors who created or significantly improved those articles. These editors are available, motivated, and capable of fixing just about anything that other editors identify as needing attention. If you make detailed suggestions, you may be gratified by quick responses to your comments.


 * Featured article review (shortcut: WP:FAR) is similar to good article reassessment. FAs do acquire problems and errors after they've passed their candidacy.


 * These reviews take place in two stages: First, a basic review with the goal of improving the article. Second, when improvements are inadequate, the article is declared a removal candidate, and editors declare whether they support keeping or removing the article's FA status; this stage is also an opportunity for editors to overcome deficiencies. Each stage typically lasts 2 to 3 weeks. Typically, a dozen or so articles are in each stage at any given time.


 * Featured list candidates (shortcut: WP:FLC). Lists are a specialized type of article (see the section about list articles). Much of the discussion on this page is about formatting, particularly tables (Chapter 14: Creating lists and tables).

Trovatore
Fraenkel 1928: Dieser interessante und bestimmte Satz, der über die damals (1874) von den transzendenten Zahlen bekannten Tatsachen weit hinausgeht, betrifft eine Menge von Zahlen, bon denen auch nur eine einzige wirklich zu bestimmnen keineswegs ganz leich ist.

Your recent modifications to Cantor's first set theory article:

Hi, Trovatore. I wish to congratulate you on your choice of a sentence to delete from the lead section of Cantor's first set theory article. Obviously, it's a high profile sentence that you disagree with. But it's also a sentence that has started to bother me: Putting dates of 2014 and 2015 in the lead section might be read as dating the research, which would be inaccurate and not relevant for a lead. However, I hadn't decided how to modify it. Your elimination of this sentence solved the problem for me. So, thank you. The other sentence you removed leads into the following sentence so it emphasizes what I am saying. I like that sentence more, but I don't think it's worth our time to argue over it.

Now we come to your statements: "There is not a disagreement. There is not a controversy.  It's such a simple question that everyone agrees.  They just phrase it differently." I think you may have a point of view that I might learn something from, so I have a few questions I'm interested in.

First, I did a lookup of "disagree" on the computer and it stated that "disagree" means "have or express a different opinion". For example, Oscar Perron and Abraham Fraenkel disagree because one of them states that Cantor's proof for the existence of transcendental numbers is a non-constructive proof while the other states that this proof is constructive. So Perron and Fraenkel have and express a different opinion. However, I do agree that it's not a controversy, which is defined as a "disagreement, typically when prolonged, public, and heated." Mostly one side ignores the other so it never gets heated. Are your definitions of "disagree" or "controversy" different from mine?

Now for your statements that I find a bit cryptic but very interesting: "It's such a simple question that everyone agrees. They just phrase it differently." Please explain why "It's such a simple question". What is the question? Also, why do you think that everyone agrees? Finally, how do they phrase it? Thank you,

@RJGray: What I mean is that the modifications of the argument to produce a specific real number are so straightforward that it is not really plausible that Perron and Fraenkel actually dispute it. Either they haven't seen it (Fraenkel I think died quite a long time ago so that's a bit of a different case), or they consider it to be enough extra argument that it's no longer really "Cantor's argument". Neither of those possibilities is very interesting, and neither constitutes a real disagreement about whether the argument gives an explicit real. The second possibility is, at most, a disagreement about what counts as "Cantor's argument", which is much less substantive than the language made it sound. --Trovatore (talk) 20:03, 5 May 2020 (UTC)

Read: User:RJGray/Translate BEFORE WRITING TO IRY-HOR.

User talk:Iry-Hor/Archive 11

This article is about Cantor's first article on infinite sets, which contains his discovery of two kinds of infinite sets: countable sets and uncountable sets. The members of a countable set, such as the fractions between 0 and 1, can be written as a sequence, for example 1/2, 1/3, 2/3, 1/4, 3/4, …. The members of an uncountable set, such as the real numbers between 0 and 1, cannot be written as a sequence. The significant developments in mathematics that came from the use of countable and uncountable sets justify the importance of this article. Also, it would be good to have another featured article on mathematics: of the 5752 featured articles, only 18 (about 0.3%) are on mathematics.

Number of featured articles:
 * Featured articles (4/5/2020: 5,741) (4/19: 5740) (4/24: 5745) (4/28: 5752)
 * English_Wikipedia (4/5/2020: 6,866) [Updated daily by a bot NOT!!]

Cantor's first set theory article Refs in lead

Translate Cantor draft

Featured_articles Cantor's Absolute Infinite Acta Mathematica

Axiom of limitation of size Von Neumann–Bernays–Gödel set theory Sandbox100 Sandbox101 Sandboxcantor1 Sandboxcantor2

Math

TODO:

MOS:MATH WikiProject Mathematics

MOS:ACCESS MOS:TABLECAPTION

Georg Cantor Cantor's diagonal argument Controversy over Cantor's theory  Axiom of dependent choice Function (set theory) GA_Review Representation theory of finite groups Lorentz group Axiom schema of replacement set theory ZFC Zermelo set theory

WP:FAC WP:Featured articles WP:Good articles WP:Mentoring for FAC Displaying a formula

File copyright tags/All

Talk:Georg Cantor's first set theory article/GA2 User_talk:D.Lazard User_talk:RJGray

WP:Verifiability, not truth WP:Good articles

https://en.wikiversity.org/wiki/WikiJournal_of_Science



FOR ARTICLE NOMINATION: User talk:Iry-Hor/Archive 11
 * Don't use this version, use one in User:RJGray/Sandboxcantor.

This article is about Cantor's first article on infinite sets, which contains his discovery of two kinds of infinite sets: countable sets and uncountable sets. The members of a countable set, such as the fractions between 0 and 1, can be written as a sequence, for example 1/2, 1/3, 2/3, 1/4, 3/4, …. The members of an uncountable set, such as the real numbers between 0 and 1, cannot be written as a sequence. The significant developments in mathematics that came from the use of countable and uncountable sets justify the importance of this article. Also, it would be good to have another featured article on mathematics: of the nearly 6000 featured articles, only 18 (about 0.3%) are on mathematics.

Number of featured articles (4/5/2020):

Wikipedia:Featured articles (5,741)

English_Wikipedia (6,866)

MOS:ACCESS MOS:TABLECAPTION

Cantor's first set theory article