Talk:Georg Cantor/Archive 2

Archive 1

Archive 2

Weyl

H. Weyl generally supported Cantor, while admitting that "at the furthest bounds of set theory, some contradictions did show up." —Preceding unsigned comment added by 86.137.170.8 (talk) 11:43, 25 July 2009 (UTC)

BBC doco

Cantor was one of four subjects (along with Boltzmann, Gödel and Turing) in the 2008 BBC doco called "Dangerous Knowledge". (see here). Not sure if this belongs in the article but I thought I'd pass it along. Manning (talk) 03:00, 25 September 2009 (UTC)

Christian or Jewish?

Georg cantor is listed both in List of Jewish scientists and philosophers and in List of Christian scientists. Which one is right? Was he Christian or Jewish? --Fibonacci 04:00, 25 Oct 2004 (UTC)

Well, according to footnotes in http://www.jinfo.org/Philosophers.html,

4. In Men of Mathematics, Eric Temple Bell described Cantor as being "of pure Jewish descent on both sides," although both parents were baptized. In a 1971 article entitled "Towards a Biography of George Cantor," the British historian of mathematics Ivor Grattan-Guinness claimed (Annals of Science 27, pp. 345-391, 1971) to be unable to find any evidence of Jewish ancestry (although he conceded that Cantor's wife, Vally Guttmann, was Jewish). However, a letter written by Georg Cantor to Paul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13, Correspondance, Gauthier-Villars, Paris, 1934, p. 306) explicitly acknowledges that Cantor's paternal grandparents were members of the Sephardic community of Copenhagen. In a recent book, The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity (Four Walls Eight Windows, New York, 2000. pp. 94, 144), Amir Aczel provides new evidence in the form of a letter, recently uncovered by Nathalie Charraud, that was written by Georg Cantor's brother Louis to their mother. This letter seems to indicate that she was also of Jewish descent, as Bell had claimed originally.

In other words, good question. --jpgordon{gab} 04:08, 25 Oct 2004 (UTC)

Depends on your definition of "Jewish", right? Some definitions would make your question have an exclusive "or". Dauben, the reigning Cantor expert, mentions the info Jpgordon gives above, but he also adds that Cantor was not Jewish "in an orthodox rabbinical sense, since his mother was a Roman Catholic". This is from Dauben's book on Cantor. However, as the whole book makes clear, Cantor was a very devout Christian, making theological researches into the nature of the Trinity even. He was also baptized a Lutheran. So Cantor should definitely be listed as a Christian. I don't know whether all this makes him Jewish --C S 03:45, Dec 5, 2004 (UTC)

My definition makes that an exclusive "or". I'm asking about his beliefs, not about his heritage. But you have already answered my question. --Fibonacci 15:41, 12 Apr 2005 (UTC)

I put him on the new and improved List of Christian thinkers in science. He could be Jewish, by culture or Jewish law of maternal inheritance, but still fit that considering his theological research. In fact I think I put one or two other Jewish scientists there as you can be both.--T. Anthony 13:28, 23 January 2006 (UTC)

Cantor was a Christian, born of Christian parents. He never referred to himself as a Jew. Moreover, he was far more enthusiastic about Christianity in his intellectual activities than was in any way required. If Jewishness is defined by beliefs and practices, Cantor was not a Jew at all.

Nevertheless, the name Cantor is very Jewish. Evidently, he descended from Jews who had converted to Christianity, a situation not all that rare then or now. If Jews are seen as a human tribal community related by descent, than he was a Jew of sorts. That his father was a stockbroker, and that he pursued abstruse mathematics are facts consistent with this. The high level of musical and literary culture in the Cantor home are consistent with this. So was Cantor's friendship with Edmund Husserl, a Jew who converted to Lutheranism in 1887. Keep in mind here that there are a few Jews who recognize Christ as the Messiah and who revere the New Testament, but who proudly describe themselves as Jews. Quite a few Jews in their day-to-day lives accept Jewish converts to Christianity as fellow Jews for certain purposes. You would not necessarily want your Jewish son to marry one but... In any event, I submit that a tribal understanding of Jewishness has more predictive power than a faith one.202.36.179.65 23:43, 16 February 2006 (UTC)

I do not see why Cantor is a specially Jewish name. "Kantor" is the title of a singer in a church service, both in Protestant churches, Catholic churches and Jewish synangogues. There are many Germanys called "Kantor", "Kenter" etc. who are not of Jewish origin. —Preceding unsigned comment added by 79.178.137.236 (talk) 12:40, 25 February 2009 (UTC)

I find the above ad nauseum discussion about the religion of his family irrelevant and racist. Are there several paragraphs in the Wiki biographies for all famous people discussing arguments for and against their inclusion in certain ethnic or religious groups? Why not? Unless it sheds light on some aspect of his beliefs or works (and CLEARLY if it is moot it does not rise to the level of inclusion here) it should not be included. boo for Wikipedia69.40.250.129 (talk) 03:00, 3 September 2010 (UTC)

Don't even try

Whether Cantor was or wasn't Jewish is a question that assumes that there are definite boundaries to the set "Jewishness." This assumption of non–continuity can neither be proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice.Lestrade (talk) 19:46, 24 February 2010 (UTC)Lestrade

What was the need for this comment? No one has brought up the question in more than a month. There is no need to go looking for trouble. --Trovatore (talk) 20:55, 24 February 2010 (UTC)

Lol I believe he was being facetious. Lestrade was simply applying relevant theorems of Cantor's study area (e.g. Zermelo-Fraenkel) and applying it to knowledgability within discussions of "Jewishness" or anything that goes into wikipedia. Genius comment, sir. —Preceding unsigned comment added by 155.31.173.86 (talk) 20:20, 7 October 2010 (UTC)

School named after him

Does it make sense to mention somewhere in the article that in Halle (the city where he died) a school is named after him (Georg-Cantor-Gymnasium)? Or maybe just link it in the "See Also" section? TFTD (talk) 07:36, 7 October 2010 (UTC)

To tell the truth, I am not very enthusiastic about doing either. Most notable historical personalities have at least one school named after them, and it's not really relevant to him, while he is of course relevant to the school.

By the way, I have reverted your edit to his death place. Our general convention is to use countries and geographical subdivisions as they existed at the time. Since Saxony-Anhalt is a post-war creation he never had anything to do with it. If you think it's very important, the section "Teacher and researcher" looks like a natural place to say something like "Halle in the province of Saxony (now Saxony-Anhalt)", but I am not sure if that wouldn't be undue weight. It looks like a borderline case to me. Hans Adler 10:14, 7 October 2010 (UTC)

Thx for the fast response and sry about the edit on the Saxony. I should have guessed that but since you changed it to Province of Saxony it's more clear and less misleading (before it stated just Saxony).

To the school thing, I totally agree with you. The intention was that the school article is marked as an Orphan, but I don't want to force anything. TFTD (talk) 09:30, 8 October 2010 (UTC)

Yes, I saw the orphan tag but neglected to mention it. I am not sure that a link from Halle, Saxony-Anhalt is appropriate either, but one could of course create something similar to de:Portal:Halle/Einrichtungen. But we would do it as a list here, and one would have to verify first that we have already enough other links for such a list. The tag is a bit silly on that article anyway, and I see it was added by an editor who was blocked last year for insisting on this kind of nonsense and vanished this April. As it disfigured the article for no discernible purpose I have simply removed it now. Hans Adler 09:52, 8 October 2010 (UTC)

Cantor's paradox

I wonder if someone who's taken the trouble to do such a nice job on this page might look at Cantor's paradox, and edit it so that it will be tied more closely to this one. (Lorenzo Traldi (talk) 15:08, 14 April 2011 (UTC))

Changes related to Cantor's 1874 article

I have rewrote the material on Cantor's 1874 article to agree with the Wikipedia article Cantor's first uncountability proof. Most of my changes are in the first two paragraphs of Georg_Cantor#Set_theory. The second paragraph uses Cantor's 1874 proofs.

In the first paragraph, I took out the unreferenced statement: "The paper, published in Crelle's Journal thanks to Dedekind's support (and despite Kronecker's opposition)… ." I have been unable to find any evidence to support this statement. On the other hand, Cantor's first uncountability proof mentions the article's "quick acceptance (only four days after submission)." Here's the timeline:

December 23, 1873: Cantor submits his article.

December 25: Cantor writes Dedekind that he has submitted a short article to Crelle's Journal containing titled On a Property of the Collection of All Real Algebraic Numbers. He says the article contains results that appear in his recent letters. Dedekind writes back and advises Cantor "to drop the restriction to the field of all real algebraic numbers." (Quote from Cantor-Dedekind correspondence, Ewald 1996, vol. 2, p. 850.) In other words, Dedekind advises Cantor to publish the stronger theorem: The collection of all algebraic numbers (both real and complex) is countable.

December 27: Cantor writes to Dedekind that Borchardt, the editor-in-chief of Crelle's Journal, has accepted the article.

By the way, Dedekind had no influence with the Berlin mathematicians, who controlled Crelle's Journal. As Ferreirós 2007, p. 185, points out: "There is some evidence that Kronecker and Kummer were angered with Dedekind after the publication of his theory of algebraic numbers in 1871." (Ferreirós then gives the evidence.) Kronecker had an equivalent theory of algebraic numbers that he did not publish until 1882. Also, Kronecker did not like Dedekind's approach since Dedekind's ideals are infinite sets of algebraic integers.

I should probably add a little on Cantor's 1878 article since it contains a proof that the set of real numbers and the set of irrational numbers have the same power. This proof can be easily modified to prove that the set of transcendental numbers has the same power as the set of reals. Before adding more, I would like to see reader's reactions (either in comments or changes) to the changes I've just made.

Also, I added an on-line link to Cantor's 1874 article. I found that the on-line link to Cantor's Gesammelte Abhandlungen mathematischen und philosophischen inhalt no longer works (at least on my computer). Does this link work for other readers?--RJGray (talk) 19:22, 29 December 2010 (UTC)

Kronecker's opposition was documented by Dauben who connects it to the weak title. Edwards has pursued the line that Kronecker did not oppose Cantor vigorously, but Dauben's research tends to refute Edwards. Tkuvho (talk) 19:52, 29 December 2010 (UTC)

Thank you for your comment. The weak title is an excellent point, and it has generated some controversy. When I was writing Cantor's first uncountability proof#Why does Cantor's article emphasize the countability of the algebraic numbers?, I realized that Wikipedia's NPOV policy required both Dauben's viewpoint (the title was caused by Kronecker's influence) and the opposing viewpoint that points to Weierstrass' influence.

For the opposing viewpoint, I used José Ferreirós' 2007 book Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought. Ferreirós' book covers the pre-history and early history of set theory as well as central parts of its modern history. It is well researched, contains lots of references, and it contains information that does not appear in Dauben's work. Ferreirós provides evidence that Weierstrass' influence was responsible for the title of Cantor's article. I'm interested in what you think about his arguments. Personally, I'm most interested in the NPOV aspect of the controversy, and letting people decide for themselves.

In any case, if Kronecker made any complaints about the 1874 article, most likely Cantor would have mentioned it — he certainly let Dedekind know about his worries concerning Kronecker and his 1878 article. Also, the article's quick appearance shows that there was no publication delay.

Thanks for pointing out Edwards' point of view. Where do Edwards arguments appear? Are they contained in his Essays in Constructive Mathematics?--RJGray (talk) 22:25, 31 December 2010 (UTC)

I don't remember anymore, sorry. I am pretty sure I found this online in a popular magazine, perhaps american math monthly. Tkuvho (talk) 15:16, 14 April 2011 (UTC)

Letter from Cantor in Russell's Autobiography

Not to bring up a potentially very silly subject that has already been beaten to death, but shouldn't the section "Cantor's ancestry" (which maybe, on the other hand, need not even exist) include that nice paragraph in Cantor's letter to Russell (the one where he says that he is not "full germain" (sic)) and proceeds to give an account more or less orthogonal to everything currently said in our page)? Feketekave (talk) 19:30, 21 December 2010 (UTC)

I am not familiar with Cantor's comment. What did he say? Tkuvho (talk) 21:13, 21 December 2010 (UTC)

Ah, here I have it. Feketekave (talk) 18:42, 27 September 2011 (UTC)

On a different matter, shouldn't we make it clearer in the lead that Wittgenstein lived quite a bit later than Cantor, at a time when Cantor's work had already become accepted by mathematicians? Some readers will be confused. Feketekave (talk) 18:52, 27 September 2011 (UTC)

Harvrefcol -> Citation

This article uses a very unusual citation templates, {{harvrefcol}}, which produces an output which is (usually) identical to {{Citation}}. (The only difference is the way that the journal volume, issue and number appear).

{{Harvrefcol}} appears in only five articles. Obscure templates such as {{harvrefcol}} tend to be poorly maintained and often have incomplete functionality. {{Citation}}, on the other hand is used in more than 56,000 articles, and is uses the same core as the {{cite *}} family of templates, which are used several million articles. This family of templates is very well maintained and well documented.

Any objections to converting {{harvrefcol}} to {{citation}}? ---- CharlesGillingham (talk) 19:10, 20 December 2010 (UTC)

I don't have any objections. Your reasoning makes sense. However, you may want to hear from some others first, if only out of courtesy to those editors responsible for this excellent article (and one of the relatively few to be a "Featured Article"). I've made only one or two insignificant additions to this page, so I'm not in the loop here. Good luck Christian Roess (talk) 20:33, 20 December 2010 (UTC)

 Done Mark Hurd (talk) 16:22, 30 September 2011 (UTC)

Poincare Criticism?

I am not sure that the statement Poincare disliked Cantor's work is correct. Reference Marcus du Satoy's recent TV Program on the story of mathematics where he claims Poincare publically supported the efforts. Also in Engines of Logic by Martin Davies claims the statement from Poincare, "a disease from which one has recovered", is apocryphal. —Preceding unsigned comment added by 82.69.53.110 (talk) 11:33, 24 September 2010 (UTC)

See the Wikipedia article on Henri Poincare. His opposition to the transfinite and the like is fully referenced in refs. 40 and 41. — Preceding unsigned comment added by 92.26.2.17 (talk) 13:57, 11 September 2012 (UTC)

Did Poincare call set theory a "grave disease"?

Some sources dispute this. DHN (talk) 09:29, 14 April 2013 (UTC)

University of Zurich or ETH?

"Youth and Studies" says University of Zurich, Infobox says alma_mater = ETH Zurich. Neither has a citation. Garymm (talk) 23:35, 1 February 2014 (UTC)

Encyclopædia Britannica says "University of Zürich". - Jochen Burghardt (talk) 23:51, 1 February 2014 (UTC)

... while Astroseti.org (the other link reachable from www-history.mcs.st-andrews.ac.uk, found via Mathematics Genealogy Project) says "Politécnico de Zurich" (which probably translates "ETH"). All of de:Georg Cantor#Leben, www.deutsche-biographie.de, and leopoldina say "University". Neither the wikipedia article on ETH Zurich nor that on University of Zurich lists Cantor among their famous students. To summarize, "University" seems to have the majority of votes. - Jochen Burghardt (talk) 00:11, 2 February 2014 (UTC)

Millerites

Does anyone know whether there's anything to the claim that the Millerites (presumably adherents of Millerism) were interested in Cantor's work? For me, a web search doesn't find much, and one of the things it does find is the same claim in this article from several years back.

So there seem to be several possibilities:

1. Just a hoax, with a persistent perpetrator.
2. Someone trying to add something he/she thinks is really true, but can't substantiate.
3. An actual tidbit about the Millerites, potentially sourceable if we found the right sources.

Now, even in case 3, I kind of doubt it's appropriate for this article, but the matter could be discussed. (Might be more appropriate at Millerism, which doesn't mention Cantor.) --Trovatore (talk) 03:11, 11 July 2014 (UTC)

I think you have expended too much time researching this already. It's clearly a little bit of promotion. I don't spend much time on wikipedia, but I've seen heaps of these little bogus promotions for various causes. So I'm sure you've seen a thousand. I imagine someone who wanted to promote millerism has done a search in wikipedia for certain keywords which appear on the Cantor page. If Cantor had been a major influence on Millerism, that could be worthy of comment on the Cantor page, but then it would surely show up on the Millerism page. But really, the Cantor page is big enough and comprehensive enough already. It doesn't need to be padded with introspective musings. Just my 2¢ worth....--Alan U. Kennington (talk) 05:36, 11 July 2014 (UTC)

Clarifying an edit summary

I wrote a rather confusing summary for this edit. I wrote:

I do not think his first proof is more complex or less elegant than the original. I think the original is more elegant.

What I actually meant was:

I do not think his first proof is more complex or less elegant than the later (1891) proof (the diagonal argument). I think the original is more elegant (than the later diagonal argument).

The original proof is applicable to all densely ordered sets that have at least two members and that have no gaps. Michael Hardy (talk) 18:18, 11 December 2014 (UTC)

Ipse dixit

In the third paragraph, there is the phrase "results now included in a standard mathematics curriculum". This is an ipse dixit. — Preceding unsigned comment added by 78.210.30.116 (talk) 10:30, 22 May 2015 (UTC)

GEORG CANTOR WAS RUSSIAN

when Wikipedia becomes nothing more than a peddler of Encyclopedia Brittanica Pop knowledge - you might as well close shop - for those who created the Georg Cantor webpage - EB is NOT a reputable source - THINKING is - try to THINK before you undo my edits - obviously you struggle with this and just fall back on EB as your "source" of the golden noodle - that is like saying the Sunday Mirror is a news source OR any other cheap trash tabloid - the man was born in St. Petersburg - he is RUSSIAN and always will be Russian - stop trying to claim him for Germany - you say you are not concerned with nationalistic sentiments - FINE - leave it alone and stop undoing the CORRECTIONS - — Preceding unsigned comment added by DigiSequenceSystem (talk • contribs) 12:14, 11 October 2015 (UTC)

CENSORSHIP

you continue to deny FACTS in favor of sensational claims supported by the beacon of TRUTH - Encyclopedia Brittanica - block me - it ONLY shows you are weak and fail to accept the FACT that Cantor was RUSSIAN - repeated attempts to claim him for Germany are driven by your own nationalistic bias - it speaks volumes about the lack of integrity of your webpage being - you cannot censor TRUTH - any idiot except you of course can see the man was Russian - Germany is well-known for stealing ideas from the Jews to make themselves look superior - this is simply another case of that and YOU in your ignorance support it - memo to idiots like YOU - Russia is capable of producing world class mathematicians too - despit the lies you peddle - Germany is not the only country that studies mathematics dum dum — Preceding unsigned comment added by 84.92.101.123 (talk) 08:46, 14 October 2015 (UTC)

The Truth will prevail - after being blocked by YOU from editing for 2 months - I return - to correct your ignorance - CANTOR WAS RUSSIAN — Preceding unsigned comment added by 94.119.64.7 (talk) 15:42, 5 December 2015 (UTC)

Remove nationality?

I wonder if it wouldn't be justified under WP:IAR to just remove any mention of nationality from the first sentence and the infobox. While WP:MOSBIO does mention "location or nationality" as an element that will usually be in the lead paragraph, MOSBIO is a guideline, and guidelines allow for exceptions.

People, please, it does not matter whether Cantor was German or Russian. Get over your stupid nationalisms. His importance is as a mathematician. --Trovatore (talk) 20:16, 5 December 2015 (UTC)

Seems like a good idea to me. Paul August ☎ 20:42, 5 December 2015 (UTC)

My hesitation is that I don't want to set a precedent for a hecklers' veto. Cantor spoke and wrote in German, not Russian (I don't know that he didn't know Russian, but I haven't seen records of him using it), he belonged to the German state church, and he lived most of his life in Germany. I think most sources are going to describe him as German.

But he didn't do his mathematics for the Tsar or the Kaiser; he did it for God, who has no nationality. --Trovatore (talk) 20:59, 5 December 2015 (UTC)

External links modified

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Article vs. Paper

I'd like to address Trovatore's recent edit in the section "‎Set theory." He justifies this edit with the statement: "it's not an article; it's a paper. Articles come in glossy magazines (and Wikipedia), not academic journals." I'm considering undoing this edit. However, I want to get some feedback first and I invite anyone justify his statement with a reliable source. In the following, I use the author guidelines from 8 mathematical journals that indicate "article" is an acceptable term and seems to be equivalent to the term "paper." I also quote from 5 Wikipedia articles. I have boldfaced the terms "article" and "paper" in the quotations below.

From Crelle's Journal:

Submission process

Set your manuscript according to the guidelines below

> Each paper should include a short but informative abstract, as well as the Mathematics Subject Classification 2010 representing the primary and secondary subjects of the article

Wikipedia American Mathematical Monthly:

The American Mathematical Monthly is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the American Mathematical Monthly fulfills a different role from that of typical mathematical research journals. The American Mathematical Monthly is the most widely read mathematics journal in the world according to records on JSTOR.

From American Mathematical Monthly:

The American Mathematical Monthly publishes articles, notes, and other features about mathematics and the profession. Its readers span a broad spectrum of mathematical interests and abilities. Authors are invited to submit articles and notes that bring interesting mathematical ideas to a wide audience of Monthly readers.

Wikipedia American Journal of Mathematics:

Fields medalist Cédric Villani has speculated that "the most famous article in its long history" may be a 1958 paper by John Nash, "Continuity of solutions of parabolic and elliptic equations".

From American Journal of Mathematics:

By submitting a manuscript, the author acknowledges that it is original and not being submitted elsewhere. The Journal's policy is to require the assignment of copyright from all contributors at the time articles are accepted for publication. Decisions concerning publication of manuscripts in the American Journal of Mathematics rest solely with the Editors.

Wikipedia Proceedings of the American Mathematical Society:

Proceedings of the American Mathematical Society is a monthly mathematics journal published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages.

From American Mathematical Society:

Where to send files for accepted papers

Links to specific instructions are available on each journal's home page.

Tracking the progress of your manuscript

Track an accepted article through the AMS journals publication stream using the Manuscript tracking system.

Making changes to articles after publication

To preserve the integrity of electronically published articles, once an individual article is electronically published but not yet in an issue, changes cannot be made in the article. The AMS policy on making changes to articles after publication provides information about submitting an errata.

Wikipedia Israel Journal of Mathematics:

Israel Journal of Mathematics is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the Bulletin of the Research Council of Israel (Section F), the journal publishes articles on all areas of mathematics.

From Israel Journal of Mathematics:

Submission of articles Papers submitted to the Israel Journal of Mathematics should be sent to iton@math.huji.ac.il, addressed to:

From Proceedings of the London Mathematical Society:

Submission of papers to the Proceedings of the London Mathematical Society

Proceedings

The Proceedings welcomes submissions of research articles where the final published length is 25 PAGES OR MORE

From International Journal of Mathematics:

Papers must be submitted with full address(es) and fax number(s) and e-mail address(es) of the author(s).

Abstracts should not be more than 300 words long.

Each article submitted must be accompanied by Mathematics Subject Classification 2000. A list of these numbers may be found in the annual Subject Index of Mathematical Reviews, published in the December issue.

I hope that this generates some discussion. RJGray (talk) 17:25, 24 August 2016 (UTC)

I just have a visceral negative reaction to "article". I find it extremely jarring in this context. It seems to downplay the originality of the work. It makes it sound like something that might show up in Time.

For work published in academic journals, it seems to me that "article" is OK for "survey articles"; that is, ones that don't report original results, but bring together what is known about something in the field. These can be very valuable service articles, but they are not original contributions. When it's an original contribution, if you don't call it a "paper", I think you're not treating it respectfully. --Trovatore (talk) 19:21, 24 August 2016 (UTC)

As far as visceral reactions go: I learned years ago that you write a "paper," and if it is printed, it becomes an "article". This explains why I prefer the term "article." For me, "article" is used only when a paper is deemed worthy of being published (so for me, the term "article" is more respectful). In fact, I had hoped that the author guidelines would prove that my view was correct. However, the guidelines of the research journals I looked at use the terms "paper" and "article" in a way indicating that they treat the terms as synonymous. For example, Crelle's Journal, which published Cantor's 1874 paper/article, starts a sentence with: "Each paper" and ends it with: "the article". The American Mathematical Society talks about "Where to send files for accepted papers" and later has "Track an accepted article". (After reading eight author guidelines, I figured I had seen enough.)

Opinions on the Net are all over the place. Here's an example that contradicts what you are saying (from Difference between research article and research paper):

What is the difference between Research Article and Research Paper?

• There is no difference as such between a research article and a research paper and both involve original research with findings.

• There is a trend to refer to term papers and academic papers written by students in colleges as research papers whereas articles submitted by scholars and scientists with their groundbreaking research are termed as research articles.

• Research articles are published in renowned scientific journals whereas papers written by students do not go to journals.

If this is a trend, I don't like it. It's taken me awhile to accept the terms "paper" and "article" as synonymous, and to realize that this can lead to clearer writing. First, I was helped by my daughter who is in grad school in soil science. When I asked her "What is the difference between the two terms?", she immediately replied "There is no difference."

Then I was helped by Gregory H. Moore's Zermelo's Axiom of Choice: Its Origins, Development, and Influence (which is a Wikipedia reliable source). On the top of page 152, Moore states: "In order to grasp Zermelo's system and its relation to the Axiom of Choice, it will be useful to re-examine Russell's article of 1906." The last paragraph of that page begins: "During July 1907, unaware of Russell's paper [1906], Zermelo completed the article [1908a] containing his axiomatization, which in Russell's terminology was a theory of limitation of size." I've read this page before and never noticed the switch from paper to article in this sentence (my mind must have read them as synonymous). It seems that Moore is taking advantage of these terms being synonymous and is avoiding repeating the word article (or paper). Moore uses the terms about equally; on page 158, he uses: papers, paper, article, paper, article, paper, article. Moore has to talk about several papers/articles at once and repeating paper or article seven times is a bit much.

My evidence derived from author guidelines and books leads me to accept "paper" and "article" as synonymous. Also, you are the first to complain about my use of "article" even though my rewrite of the old "Cantor's first uncountability proof" article starting appearing in May 5, 2009‎. This seems to imply that there is an implicit consensus that "article" is an acceptable term. Also, I can point to a number of Wikipedia articles that contain both the terms "paper" and "article" in them.

So I disagree that my use of the word "article" is disrespectful. I think very highly of Cantor's article and have been trying to correct the notion that Cantor gave a non-constructive existence proof when he should be given credit for presenting his work constructively.

I find "paper" and "article" equally acceptable, and will refrain from imposing my old preference for "article." In major rewrites and in new articles, I will feel free to use either term. Because of this, I will not undo your current edit. The original text of December 29, 2010 used the word "paper" twice and I was only making small corrections to the paragraph they were in. (I completely rewrote the next paragraph to give Cantor's constructive proof of the existence of transcendentals, which replaced a paragraph that had him presenting a non-constructive proof.) So I regard your edit as correcting my old edit. If I had done the edit today, I would have respected the original editor's use of the term "paper." --RJGray (talk) 18:47, 27 August 2016 (UTC)

Absolute infinite, well-ordering theorem, and paradoxes (rewrite of "Paradoxes of set theory")

I have rewritten the "Paradoxes of set theory" section. I thank the editor who referenced Hallett's book in the original section. This book was a great starting point. I also thank the editors of the Burali-Forti paradox and Cantor's paradox articles who referenced the Moore and Moore & Garciadiego articles. These articles have been very helpful. What follows is an explanation of my rewrite and how I went about it. My explanation is a bit long because I came across many interesting facts during my research for the rewrite. I will use some of this material to rewrite the Wikipedia article "Absolute infinite," which currently has a maintenance template in it. The following statement by Moore affected the way I wrote the section:

… later opinions have been influenced so strongly by the traumatic view of the paradoxes which Russell set forth in The Principles of Mathematics [1903]. One should observe, first of all, that Cantor exhibited no alarm over the state of set theory in his letter [to Hilbert]—in sharp contrast to Gottlob Frege's dismay upon learning in 1902 of Russell's paradox. What Cantor remarked was merely that certain multitudes are, in effect, too large to be considered as unities (or sets) and so are termed absolutely infinite. Significantly, he retained such absolutely infinite, or inconsistent, multitudes and even employed them in the proof of the Aleph Theorem that he sent to Dedekind. Thus Cantor did not treat these apparent difficulties as paradoxes or contradictions, but as tools with which to fashion new mathematical discoveries. (Moore, Gregory H. (1982), Zermelo's Axiom of Choice: Its Origins, Development & Influence, p. 53.)

Hence, caution is needed when reading accounts of the paradoxes of set theory. Because of this, I have tried to make sure that every sentence in this Wikipedia section states a fact and not just someone's opinion. I've used Hallett's book, and Moore's book and articles because they reference primary sources, such as letters. Also, Moore (starting in 1981) seems to have done the most detailed analyses of the paradoxes. For those interested in the paradoxes, I recommend the Moore and Moore & Garciadiego articles, which are available free online.

The new section covers Cantor's ideas, the paradoxes that Russell discovered, and two mathematical solutions to the paradoxes: (1) Zermelo's 1908 axiomatic solution that restricts the formation of sets, and (2) von Neumann's axiomatic solution using proper classes (classes that are not sets). There is a third solution, Russell's theory of types, but it is less relevant to this article and is more complex to explain.

Cantor not only knew about the contradictions that occur by assuming certain multiplicities are sets, but he also considered the problem of proving consistency. In an 1899 letter to Dedekind: "Cantor declared that one could not even demonstrate the consistency of every finite set. Such consistency was 'a simple indemonstrable truth,' which he termed the Axiom of Arithmetic [Cantor 1932, 447–448]. In a similar fashion he regarded the consistency of each aleph as an indemonstrable truth, which he named the Axiom of Extended Transfinite Arithmetic." (Moore 1982, p. 54.) Gödel's work proved that the consistency of a finite set theory that supports Peano arithmetic cannot be proved within the theory.

Cantor is clearer in his letters than in his articles. Concerning his 1883 definition of a set: In a 1907 letter to Grace Chisholm Young, Cantor stated that when he wrote his 1883 Grundlagen, he saw clearly that the ordinals form an inconsistent multiplicity rather than a set. He pointed out that in the endnotes of his Grundlagen: "I said explicitly that I designate as "sets" only those multiplicities that can be conceived as unities, i. e. objects, …." (Moore and Garciadiego 1981, p. 342.) Although Cantor stated in the Grundlagen that the ordinals form an absolutely infinite sequence, he did not explicitly state that conceiving all the ordinals as a unity leads to a contradiction.

Concerning his 1895 definition: By a "set" we are to understand any collection into a whole M of definite and separate objects m of our intuition or our thought. (Cantor 1955, p. 85. I've used "set" rather than the old term "aggregate.") It has been claimed that Cantor's definition leads to "naive set theory." However, in an 1897 letter to Hilbert, it is clear that Cantor did not intend his definition to be interpreted in this way:

I say of a set that it can be regarded as comprehensible … if it is possible (as is the case with finite sets) to conceive of all its elements as a totality without implying a contradiction. … For that reason I also defined the term "set" at the very beginning of the first part of my paper [his 1895 paper whose "set" definition is given above] … as a collection (meaning either finite or transfinite). But a collection is only possible if it is possible to unite it." (Purkert, Walter (1989), "Cantor's Views on the Foundations of Mathematics", in Rowe, David E.; McCleary, John (eds.) (eds.), The History of Modern Mathematics, Volume 1, Academic Press, p. 61 {{citation}}: |editor-first2= has generic name (help).)

I did not mention "limitation of size" because Cantor viewed the difference between the transfinite and the absolute infinite originally in terms of increasable/unincreasable and later in terms of consistent/inconsistent. Hallett says that the limitation of size hypothesis (all contradictory collections are too big) is a "spiritual descendant of Cantor's way of thinking represented in his 1899 correspondence." (Hallett 1986, p. 176.) Hallett is interested in the development of ideas and is looking for possible antecedents. However, this Wikipedia section deals in history and Cantor did not take the step of formulating "limitation of size." Hallett goes on to say: "But in published form LSH [limitation of size hypothesis] and its use as a starting point for building a contradiction-free set theory stems from Russell and Jourdain." (Hallett 1986, p. 176.)

I find Zermelo's handling of the paradoxes interesting. In his 1908 set theory article, he stated that his axioms exclude the known paradoxes. In his 1930 article on models of set theory, he gave a new explanation of the paradoxes. He postulated that there exists an unbounded sequence of strongly inaccessible cardinals κ, defined a sequence of models V_κ satisfying von Neumann's axiom, and explained why the paradoxes are only apparent "contradictions":

Scientific reactionaries and anti-mathematicians have so eagerly and lovingly appealed to the 'ultrafinite antinomies' in their struggle against set theory. But these are only apparent 'contradictions', and depend solely on confusing set theory itself, which is not categorically determined by its axioms, with individual models representing it. What appears as a 'ultrafinite non- or super-set' in one model is, in the succeeding model, a perfectly good, valid set with a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain [model]. (Ewald, William B. (1996), From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2, p. 1233.)

To get a feeling for Cantor's absolute infinite, imagine being inside one of Zermelo's models V_κ where κ is a strongly inaccessible cardinal. There is no bound on the ordinals of the model. Also, the class of all ordinals cannot be increased in magnitude since there are no larger ordinals to add to it. Therefore, this class is unincreasable, which is a feature of Cantor's absolute. Looking at the model from the outside, this class is the set of ordinals < κ. Of course, κ is not in the model.

I learned a lot by doing research for this section. I hope that readers will finding my rewrite informative and interesting. RJGray (talk) 17:27, 18 September 2016 (UTC)

Oedipus complex?

Historiography says Bell describes Cantor's relationship with his father as Oedipal. It's perhaps worth noticing that Bell avoids Freudian jargon, and indeed Men of Math Chapter 1 says that the only mathematician he considers who would interest a Freudian is Pascal. Also, Bell quotes a letter from Cantor to Cantor's father to support the statement that Cantor had a servile attitude toward his father, so presumably Bell wasn't simply making up that part. — Preceding unsigned comment added by 209.159.232.121 (talk) 19:39, 6 May 2017 (UTC)

That jumped to me too. I think that can be deleted. After all, it does not explain why he had an Oedipal relation with his father, and that behavior is more common related with a mother not a father.

More detail

I want more detail about Cantor's ideas of the authorship of the works usually credited to Shakespeare. — Preceding unsigned comment added by 2A02:C7D:B300:C700:B5F8:9857:D22D:26B (talk) 15:18, 20 May 2017 (UTC)

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Context for a statement in intro

There is a statement in intro, attributed to Wittgenstein, which is misleading without proper context. That statement should be inserted in Wittgenstein article where I see that it is missing.--5.2.200.163 (talk) 13:02, 20 June 2018 (UTC)

Platonist

According to Morris Kline Cantor said of himself that he was a Platonist [Kline (1980) "Mathematics: The Loss of Certainty", p. 203]. Perhaps this can be added to the article? 2A02:A03F:3EE0:DA00:A511:B872:71BA:5365 (talk) 09:24, 29 September 2018 (UTC)

Reversion of recent edit

I (and another editor) have together effectively undone this edit. The original text said:

In 1883, Cantor divided the infinite into the transfinite and the absolute. The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α + 1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it. In 1883, Cantor also introduced the well-ordering principle "every set can be well-ordered" and stated that it is a "law of thought."

References

But was changed to read (I've bolded the new text):

In 1883, Cantor divided the infinite into three categories: the potential infinite, the transfinite and the absolute The potential infinite is represented by a quantity that can indefinitely grow, tending to the infinite as its own mathematical limit, and differs from the actual infinite in that only the second already exists in the space-time, whereas a mathematical entity can reach its reality in a thinking mind (such as the man, or God the Creator), not always being corresponded by a measurable object, and with the unique condition not to originate any rational paradox.

The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α + 1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it. In 1883, Cantor also introduced the well-ordering principle "every set can be well-ordered" and stated that it is a "law of thought.".

Among transfinite numbers, Cantor distinguished aleph numbers: aleph-zero (the infinite discrete set of the integer numbers), aleph-one (the continous infinite set of the real numbers, or in geometrics the set of points of a segment), and more others.

References

I'm concerned that this may misrepresent what Cantor wrote. Unfortunately I'm unable to verify the accuracy of the new text. However, at best, it's poorly written, and even if true perhaps not needed? Comments? Paul August ☎ 19:27, 3 April 2019 (UTC)

It's sourced to L'infinito by Antonino Zichichi, which I think I might enjoy reading, but looks to be a popular account rather than serious scholarship (the subtitle is la più grande conquista della logica matematica raccontata come una favola, "the greatest conquest of mathematical logic, told as a fable"). I also am not aware that Cantor had much to say about the "potential infinite", though that doesn't prove much. --Trovatore (talk) 19:39, 3 April 2019 (UTC)

(Actually I should amend that. The subtitle I gave is for the edition I found on Amazon, from 1994. There does seem to be a later version, from 2004, with the subtitle as given, "the adventure of an extraordinary idea" -- https://books.google.com/books?id=8GgqAAAACAAJ&dq=isbn:9788851522124&hl=en&sa=X&ved=0ahUKEwjSuazX17ThAhXVIDQIHefNAIMQ6wEILTAA.) --Trovatore (talk) 19:46, 3 April 2019 (UTC)

Kantor's ancestry

This section is two pages long - out of 11 pages of the whole biography. I think the section falls under WP:UNDUE for the whole question of Cantor's ancestry is quite marginal to his life and work. The same way, I do not see a reason for the Cantor's brother Louis message sent to his mother written in German.--Dishonesty Test (talk) 11:40, 3 July 2019 (UTC)

I might be wrong, but ...

As to the religion, I see some whitewash in the Philosophy, religion, literature and Cantor's mathematics section, By the way, I do not see anything under this section related to the literature.

From L.C. YOUNG: LECTURES ON THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY, Publisher: Philadelphia, PA : W.B. Saunders, 1969. page 103 we read:

The story, as well as the work, of Georg Cantor deserves to be remembered. It is indeed a sad one. In his own time, he was declared insane after he had published a pamphlet claiming ( as many clergymen now claim) that Christianity is much more beautiful without the virgin birth. Coming after his revolutionary mathematical concepts, this was the last straw.  His bitter enemies saw to it that he was kept in one of the terrible institutions of those days for nearly 20 years, until he died.

From T. L. Casey, G. Van Ingen, C. L. Poor, E. O. Hovey, R. W. Tower (editors): Annals of the New York Academy of Sciences, Volume 321, New York Academy of Sciences, 1979 – page 44 we read:

On April 5, 1905, Cantor wrote to Grace Chisholm Young, describing as he had to Jourdain the results of his being "hermetically secluded" in the Halle Nervenklinik forfive and a half months: "As you know, I had been hermetically secluded 5 1/2 months (from 17. Sept. to 1. March) from the world, except few visits from my family."

From L. Young: Mathematicians and Their Times, Volume 48 of North-Holland Mathematics Studies, Elsevier, 2011, page 232 we read:

Of the cause and treatment of Cantor’s depressions, I can only speak with considerable reserve. ...
In the times of rule of thumb authoritarianism, regardless of truth, even the best mathematicians can be quarrelsome and intolerant, especially to someone who breaks not only with logstanding mathematical traditions but established beliefs outside mathematics, such as the authorship Shakespeare’s plays and the Immaculate Conception. How would this hostility of a number of fellow mathematicians affect a man like Georg Cantor, whose skin hardly possessed insesitivity of a rhinoceros? ...
Cantor complained bitterly, in a letter to my mother about 1905, of a "conspiracy to have him committed".

The bottom line is the L. C. Young's "Of the cause and treatment of Cantor’s depressions, I can only speak with considerable reserve." Then why on the Earth a mathematician should talk to some theologians and send a letter to Roman Pope about the nature of infinity? My educated guess - his confinement in the lunatic asylum was a sort of the Roman Catholic Church Inquistion punishment for his stance about the virgin birth.--Dishonesty Test (talk) 15:10, 3 July 2019 (UTC)

Cantor's father

Regarding Cantor's father, there are at least three different spellings of his name:

• Georg Waldemar
• George Woldemar
• Georg Woldemar

Please decide which variant is correct. Herpesklaus (talk) 20:33, 18 August 2019 (UTC)

Picture is mirrored?

I was wondering if there is a reason, why the picture in the English version is the mirror picture to the German version? I think it would make sense to have some kind of consistency. — Preceding unsigned comment added by 2003:C8:9F04:2F00:E123:63FE:7D64:521F (talk) 21:27, 19 December 2019 (UTC)

Good observation. It would be interesting to find the source of these images and to see what is original. —Kusma (t·c) 21:52, 19 December 2019 (UTC)

I downloaded both, and I do not believe they are the same photo. Some of the differences may be just post-processing, but I think Cantor's head is just slightly more tilted in the de.wiki version. He is wearing the same (or a very similar) outfit and appears to be the same age. Maybe it was two photos from the same photo session, one a profile from the right and the other from the left? --Trovatore (talk) 21:59, 19 December 2019 (UTC)

After flipping one of the photos in ImageMagick and flipping back and forth between them, I take it back. I think they are the same photo. The de.wiki one is higher general quality. I suspect the en.wiki one has been rotated slightly to make Cantor appear more vertical. Also, this is a little hard to be sure, but the en.wiki one seems to show his coat with the right lapel slightly farther from his body, implying the buttonholes are on the right and the buttons on the left, which is not the way I am used to men's coats being constructed. So maybe we should use the de.wiki one, which seems likely to be more accurate and in any case just looks better. --Trovatore (talk) 22:20, 19 December 2019 (UTC)

Interestingly, the source given for the left-looking picture commons:File:Georg Cantor2.jpg on Commons is this, where Cantor looks to the right. Materialscientist, do you remember what happened there? —Kusma (t·c) 11:16, 20 December 2019 (UTC)

I wonder if it had something to do with this notion that portraits should be looking towards the text? Personally I don't think that's a good enough reason to mirror a photo. Faces are never perfectly symmetrical, so mirroring the picture, in effect, gives a false representation of the person's appearance. --Trovatore (talk) 21:26, 20 December 2019 (UTC)

I boldly subbed in the superior image, which also appears to be in the original orientation. Not claiming this is the end of the discussion, but it seems like a clear improvement to me. --Trovatore (talk) 01:42, 21 December 2019 (UTC)