Talk:Georg Cantor

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Article vs. Paper[edit]

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, addressed to:

From Proceedings of the London Mathematical Society:

Submission of papers to the Proceedings of the London Mathematical Society


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")[edit]

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.), The History of Modern Mathematics, Volume 1, Academic Press, p. 61  .)

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 Vκ 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 Vκ 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?[edit]

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 (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[edit]

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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