Talk:Controversy over Cantor's theory

Cantor Anti-Diagonal Argument — Clarifying Determinateness and Consistency in Knowledgeful Mathematical Discourse

Perhaps my unfinished manuscript "Cantor Anti-Diagonal Argument -- Clarifying Determinateness and Consistency in Knowledgeful Mathematical Discourse" would be useful now to those interested in understanding Cantor anti-diagonal argument. I was hoping to submit it to the Bulletin of Symbolic Logic this year. Unfortunately, since 1 January 2008, I have been suffering from recurring extremely blurred vision due to frequent “exploding optical nerves” brought on by my diabetes (I can’t afford laser eye surgery) and I had only about 20 productive days in the last 8 months. At this rate, it would take me a long while to finish my paper or may not be able to complete it if I go permanently blind soon. I just hope my endeavors to clarify mathematical infinity and modern logic would reach the next (if not the present) generations of mathematicians, philosophers, and logicians. [BenCawaling@Yahoo.com] BenCawaling (talk) 08:14, 4 September 2008 (UTC)

A Few Computer Scientists Know Think Cantor Diagonal is Wrong

While most "main stream" mathematicians accept Cantor's Diagonal Proof, it's easily understood to be false by a few computer scientists.

How many real numbers are there? Simple. As many as there are computer programs capable of generating the nth digit of a real number when asked. The total number of finite computer programs is countable, and can be listed. Every real number is generated by an algorithm on the list.

So, why doesn't Cantor's Diagonal Argument disprove my simple list? My list of all algorithms contains invalid algorithms that will "hang" and never finish calculating the nth digit when asked. The Halting Problem shows that we cannot ever remove all the algorithms that hang from the list. Therefore, Cantor's algorithm hangs! QED.WaywardGeek (talk) 20:02, 2 June 2009 (UTC)

Interesting. I am not sold about the claim that all real numbers can be described. Certainly all finitely describable real numbers are countable since text is countable. Thus it is impossible to prove a real number is missing from such a list by finishing a description of it since any finitely describable real number would be in the list. Of course if one could prove the existence of any finitely indescribable real number, then such a number would not be in the list by definition.

QED.meadbert (talk) 20:02, 2 June 2009 (UTC)

WaywardGeek seems to be claiming that all real numbers are computable (not just describable), which is incorrect. But WaywardGeek is certainly right that the set of computable real numbers is countable, and that Cantor's diagonal argument does not apply to this set (because the diagonal number needn't be computable). --Zundark (talk) 21:10, 2 June 2009 (UTC)

You are absolutely correct. I would be classified a constructionist, I suppose.WaywardGeek (talk) 22:07, 2 June 2009 (UTC)

Cantor's theorem says "there is no surjection from the natural numbers to the reals". If you decline to recognize non-computable reals, then for coherency's sake you should also decline to recognize non-computable surjections. And now, guess what: You agree with the theorem.

I've indulged myself briefly here, going off-topic in response to an off-topic post, but please be aware that this is not what talk pages are intended for. Per WP:TALK, talk pages are meant to discuss improvements to the corresponding article, not to discuss their subject matter. If you want to argue about this, there's always Usenet, or a hundred other possibilities. --Trovatore (talk) 00:43, 3 June 2009 (UTC)

I think the article could be improved by pointing out that the only real numbers not included on the countable list of computable numbers are bazaar numbers who's value cannot ever be known (and thus never listed). These definable, but unknowable values do not impact our universe in knowable ways.WaywardGeek (talk) 14:29, 13 June 2009 (UTC)

Thank goodness the article does not include such nonsense! The statement that certain real numbers do not "impact our universe in knowable ways", while (apparently) other reals do, is just ill-defined silliness. Phiwum (talk) 21:12, 20 November 2009 (UTC)

WaywardGeek's comment is NOT nonsense. The set of all computable numbers is "countable". Any number that is not computable, you cannot point to it, you cannot show it to me, you cannot name it. It is a form of religious belief to say that something you cannot see, touch, depict, or name is somehow "real". It is like bringing the concept of God into mathematics. And to call all the writings of some of the most brilliant mathematicians in intuitionism and constructivism "nonsense" is not very bright. Just say you disagree.Worldrimroamer (talk) 08:12, 11 May 2010 (UTC)

While non-computable numbers have no explicit usage in any forms of numerical analysis - they are crucial to the formalism behind measure theory, which is a cornerstone of modern analysis. Please don't be arrogant enough to claim a mathematical entity is useless just because you don't have the basis needed to fathom it's importance and beauty. Also, please note that claiming a definition to be "useless" doesn't falsify it. You're merely stating that while there are uncountably many numbers we should only care about a countable subset of them. Subjective approaches were never useful in mathematics.

You are mistaken. There are uncomputable numbers that are definable. Chaitin's constant is perhaps the best known example. In any case, what WaywardGeek wrote was nonsense, regardless of the respectability of intuitionism and constructivism. The comment about "impact[ing] our universe in knowable ways" is vague and, near as I can see, meaningless. It also does not resemble any constructivism or intuitionism I've seen. Phiwum (talk) 13:43, 11 May 2010 (UTC)

Please don't paint all computer scientists with the same brush. Most of us do believe Cantor's theorem. — Preceding unsigned comment added by 192.17.205.74 (talk) 19:57, 20 November 2009

I imagine what WaywardGeek meant to say is that of the uncountably many real numbers only countably many are definable. I can understand if someone takes this as an indication that mathematicians have run off in the wrong direction. This paradox (or at least something closely related) has once bothered some great mathematicians. See Skolem's paradox. Hans Adler 21:34, 20 November 2009 (UTC

Cantor's proof of the existence of transfinite numbers by his second diagonal argument does not proof what it pretends to proof, it only falsifies the initial supposition that it is possible to list all real numbers between 0 and 1 in a square matrix, which is necessary to the construction of a number, that is not in this list (diagonal argument). As all numbers with n digits can not be listed in square matrix but only in a list with 10^n rows (10^n is read 10 to the nth power) and for such a list, there is no way to construct a number, that isn't in this list. The same is true for n+1 and so on ad infinitum by the way of mathematical induction. (in German: http://www.homepage.bluewin.ch/textarchiv/Logik/Infinitus.html)

This argument is irrelevant to this article, since it is unpublished original research. It is also errant nonsense, but an argument to that effect is irrelevant here. This page is for discussion of improvements to this article. Phiwum (talk) 15:52, 26 December 2009 (UTC)

I think his/her point is that although one can show that there "exists" a number that is not in the countable list of all rationals, one cannot construct it. One cannot see it or name it or even point approximately to it, not even if you continued diagonalizing for all eternity. I think the poster is simply espousing mathematical constructivism, which is certainly not "nonsense".

My (relevant-to-this-article) point is that is that the empiricist and constructivist comments in this article should be left as they are. This is a major issue in mathematics and the philosophy thereof. Worldrimroamer (talk) 08:28, 11 May 2010 (UTC)

Most constructivists accept that pi and the square root of two are computable numbers, although they are not rational. I think your view of constructivism is a bit off. In any case, if this page should exist at all, then of course intuitionism and constructivism should be represented here — with appropriate citations to reliable sources. Half-thought original arguments, on the other hand, have no place in the article. Phiwum (talk) 13:48, 11 May 2010 (UTC)

I think that'd be arrant nonsense? Errant nonsense sallies forth, righting all wrongs.... --Trovatore (talk) 10:57, 27 December 2009 (UTC)

Yes, thanks.Phiwum (talk) 20:09, 27 December 2009 (UTC)

Trying to compare mathematics to religion is such a crank move. TheZelos (talk) 14:28, 5 April 2016 (UTC)

Removal of Bishop quote

I removed this quote from the section about Hume:

"Classical mathematics concerns itself with operations that can be carried out by God… Mathematics belongs to man, not to God… When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself." (Errett Bishop (1967))

First of all, I'm looking at the 1967 book, and the second sentence occurs on page 2, but the first sentence doesn't seem be anywhere before it. Second, Bishop does not mention Cantor in this book on or before the page on which the quote occurs. According to the index, he mentions Cantor only on page 25:

"Theorem 1 is the famous theorem of Cantor, that the real numbers are uncountable. The proof is essentially Cantor's 'diagonal' proof. Both Cantor's theorem and his method of proof are of great importance."

Bishop praised both Cantor's theorem and Cantor's method of proof, in the only context where Bishop mentioned Cantor, which is entirely separate from the statement about God (where Bishop was saying positive things about Kant and Kronecker). Well, but what about Hume? Bishop's 1967 book has nothing to say about Hume, according to the index. Bishop's alleged role in this particular alleged controversy appears to have no documentation. Anyway, the first quote does not belong in the article as it stands. Maybe the one on page 25 does, but I doubt it; anyway it would seem to put Bishop on the other side of the controversy (i.e., supporting Cantor), if anywhere.66.245.43.17 (talk) 23:13, 15 October 2009 (UTC)

Time frame; synthesis

What is the time frame for this controversy? Cantor died in 1918. Kronocker died in 1891. Yet, the article quotes people in the late 20th century or even the beginning of the 21st century, as though they were all on two opposing basketball teams or political parties. Mathematicians probably can't be neatly categorized into "Cantorians" and "Anti-Cantorians". This idea of a controversy lasting more than a century seems to be a vague abstraction. The involvement of various mathematicians appears to be exaggerated by quotations taken out of context which may not in reality have been addressing the subject of this article. It's very interesting material, and certainly everything is related to everything else if you think about it hard enough, but there are problems with "original research" and "synthesis" in Wikipedia articles. The fact that this particular article is about "controversy" means it needs to be especially focused to avoid synthesis. I suggest that the editors of this article define its purpose and scope more precisely and narrowly. 66.245.43.17 (talk) 21:57, 16 October 2009 (UTC)

Zenkin Gets No Mention

Zenkin, A. (2004) 'Logic Of Actual Infinity And G. Cantor's Diagonal Proof Of The Uncountability Of The Continuum', The Review of Modern Logic 9, 3&4, pp.27-82.

http://projecteuclid.org/euclid.rml/1203431978 —Preceding unsigned comment added by 71.251.170.159 (talk) 21:07, 23 March 2010 (UTC)

Zenkin

An anonymous IP has repeatedly added references to Alexander Zenkin's articles. I've reverted a few of these. From what I can see, Zenkin has published some of his "anti-Cantorian" writings, though I'm unfamiliar with the journals and don't know their reputations. As far as I can see, his articles are uncited by anyone aside from Zenkin.

Personally, I believe that Zenkin is a well-educated crank, but the fact seems to be that he has some publications of his research — in my personal opinion, these articles are very dubious, but my personal opinion counts for bupkiss.

The latest addition of a Zenkin reference is as a citation to the following text:

Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world, for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence.

I think Zenkin's article does indeed say this (but I haven't a copy at hand). Regardless, I'm not sure that he's the best source for this claim. Is he a well-respected historian of mathematics?

I have not reverted the citation, because it does at least seem relevant to the text. I'd like to hear from others about whether this is an appropriate citation. Phiwum (talk) 18:37, 25 March 2010 (UTC)

For someone not even aware of the inappropriate citation of Hodges in the references when citation was needed for a quote from the same paper in the second paragraph, it leads me to question your expertise on the subject. A minimal amount of investigation with the link given would have revealed that The Review of Modern Logic is hosted by Project Euclid which "was developed and deployed by the Cornell University Library and is jointly managed by Cornell and the Duke University Press." The founding editor of the journal is, historian of logic, Dr. Irving H. Anellis, and its editorial and advisory board can be found here:

http://projecteuclid.org/euclid.rml/1203431974

It seems strange that you "haven't a copy at hand" given that the full article is open access. Furthermore, the nature of the citation becomes quite clear when actually reading the reference in regard to clarifying potential versus actual infinity and the strange disappearance of the debate from most texts.

Indeed The Review of Modern Logic is interested in all aspects of symbolic logic, including foundations, the foundations of mathematics, and set theory, though it is not particularly interested in publishing papers devoted to the technical development of pre-existing logical or axiomatic systems. Accordingly, The Review of Modern Logic is interested in publishing papers in the history of symbolic logic, in the philosophy of symbolic logic...

The Review of Modern Logic is interested in publishing surveys of developments in symbolic logic, such as a survey of alternative set theories; translations of works in symbolic logic, such as a translation into English from the original German of one of Cantor's papers on set theory; and source materials in symbolic logic, such as a letter from Cantor to Hilbert."

See: http://journalseek.net/cgi-bin/journalseek/journalsearch.cgi?field=issn&query=1047-5982

Reading the purpose of this journal leads one to believe that the citation is quite warranted in the context of an encyclopedia entry.

The author is mostly published in the Russian language, which hardly makes one a crank. I can name someone that you know who has an axe to grind. I think that “self-absorbed academics” always have an axe to grind. I would rather call them pathological narcissists, but so goes a whole generation. By the way, say hello to Tyler Durden for me. —Preceding unsigned comment added by 71.126.11.30 (talk) 02:00, 26 March 2010 (UTC)

This is a very long delayed response, since I only now read your message. I should say that I've never claimed to be an expert on the historical debate regarding infinite sets. I'm not an expert. I haven't read the Hodges citation. I don't claim to have a broad view of the issues.

And I don't know who Tyler Durden is. Sorry. Phiwum (talk) 03:38, 16 August 2015 (UTC)

Okay, WP taught me who Tyler Durden is, but I don't get your point. Phiwum (talk) 03:40, 16 August 2015 (UTC)

Unmitigated nonsense

I found the following bullet point in the article:

That there exists at least one infinite set of things, usually identified with the set of all finite whole numbers or "natural numbers". This assumption (not formally specified by Cantor) is captured in formal set theory by the axiom of infinity. This assumption allows us to prove, together with Cantor's theorem, that there exists at least one set that cannot be correlated

one-to-one with all its subsets. It does not prove, however, that there in fact exists any set corresponding to "all the subsets".

Since the Cantor's theormem referred to in this bullet point is the one that points out that no set can be correlated one-to-one with all its subsets, thsat amounts to saying that it is the axiom of infinity that allows us to prove that at least one set exists. I don't think I need the axiom of infinity to prove the existence of the singleton set containing the number 0, or the existence of the empty set, or indeed of any recursive finite set. So the bullet is just nonsense. So incidentally is the idea that Cantor was in any way out of touch with mainstream mathematicians in wanting to allow infinite collections. Two years earlier Peano's axioms required the natural numbers to be closed under addition, so there clearly was not a finite number of natural numbers. Something over two thousand years earlier Euclid (or was it Pythogoras) had proved that there were an infinite number of primes. Peanos work was based on Dedekinds, which clearly recognised that there were an infinity of natural numbers. The whole idea that Cantor's work created enormous controversy is unmitigated nonsense too - what it provoked was a few snide comments from mathematicians looking for a good sound-bite - the record for starting the biggest brouhaha ever in mathematics belongs very clearly to David Hilbert, not to Cantor. The article has carried a citation needed mark on the biggest controversy statemenmt for a long time - too long a time. I think that statement needs removing now. I've modified it to make it clear that any Cantor controversy was trivial compared to what Hilbert caused, but that isn't the right treatment - the whol;e article needs reqritubg to get rid of this fictional controversy story. 79.153.237.73 (talk) 22:29, 29 March 2010 (UTC)

Oh dear, this bullshit was added by Peter Damian in November 2005! [1] Congratulations for being the first mathematician who read this paragraph! But I am not going to touch this article, because I simply don't know what to do with it other than propose it for deletion. Hans Adler 23:01, 29 March 2010 (UTC)

It's somewhat mitigated nonsense. Everything would be correct if you replaced the third sentence with this allows us to prove that there is at least one set that can be correlated one-one with one of its proper subsets. It's true that the passage would no longer be particularly relevant, but it's an understandable confusion from a novice's point of view. --Trovatore (talk) 00:35, 1 April 2010 (UTC)

Cohen has closed the controversy by own proof

Well described here: http://en.wikipedia.org/wiki/Paul_Cohen_(mathematician)#On_the_Continuum_Hypothesis —Preceding unsigned comment added by 46.98.1.144 (talk) 04:32, 11 September 2010 (UTC)

lede tone

The tone of the introduction leaves room for improvement. Tkuvho (talk) 16:18, 13 February 2011 (UTC)

Hilbert's non-constructive proof of the Hilbert basis theorem was not "a number of decades later", but rather a year earlier. Tkuvho (talk) 16:21, 13 February 2011 (UTC)

The whole article is awful (this is a common trait for articles about math that have the word "controversy" in the title).

I don't know what this article is supposed to be about:

• Cantor's diagonal argument, or
• Set theory in general

If it's the former, I think it would be beast to drastically cut down the article, move it to a section of Cantor's diagonal argument entitled "Controversy" or "reception", and redirect this page there.

If it's the latter, then I think the article should be moved and/or merged to something that makes it more clear that the argument is not with "Cantor's" theory, but with set theory in general. The actual theories studied by Cantor, like Principia mathematica, is primarily historical now. Mathematicians use ZFC set theory, not Cantor set theory. — Carl (CBM · talk)

Is it perhaps about the history of the reception of Cantor's set theory, with the prime example being the diagonalisation argument? Certainly the objections cited here go beyond diagonalisation. Perhaps we can rename the page "History, etc." to avoid the inflammatory term. Tkuvho (talk) 16:52, 13 February 2011 (UTC)

I asked Trovatore on his user talk page to see what he thinks about Criticism of set theory. The issue is that the more philosophical criticisms about set theory are not about the diagonal argument at all; criticisms of the diagonal argument itself are generally limited to cranks these days.

The phrasing "Cantor's theory" is needlessly personal: the criticisms that are relevant today apply to Zermelo-Fraenkel set theory, or to set theory in general, but there's nothing about Cantor's theory in particular that's relevant. — Carl (CBM · talk) 17:54, 13 February 2011 (UTC)

I think the odd title is due to the history of the article. It was created by an "anti-Cantorian" and originally consisted of original arguments about how set theory took a wrong turn with Cantor.

I agree that the title is odd now that the article isn't about Cantor per se. Phiwum (talk) 22:11, 13 February 2011 (UTC)

BTW your point about Hilbert's basis theorem is completely apropos. — Carl (CBM · talk) 17:55, 13 February 2011 (UTC)

"Criticism of set theory" is fine though I worry about losing the historical focus. If we retain Cantor's name, as in "Reception of Cantorian set theory", this would naturally create an incentive to give priority to history, which sometimes gets lost in the shuffle when mathematicians work on a page. Your list at Trovatore's page is terrific. Feferman's criticism is currently not represented at all, while being perhaps the most potent. Tkuvho (talk) 20:02, 13 February 2011 (UTC)

I hear what you're saying about the history, but the difficulty is that if we want to focus specifically on "Cantorian" set theory then this will make us skip most of the modern (say post-1940) criticisms, since these will all be written about ZFC, not about Cantor's historical theories. On the other hand, I think that if we have a general article on criticisms of set theory, it would be natural to devote a significant amount of space to the "early" criticisms.

I have seen some quotes from Feferman in previous versions of this page, and including his opinion makes sense. I guess I was thinking of it as a more modern form of predicativism when I made my list. I'm sure there are also other things that could be included. The difficulty will probably be to find a good overall structure for the article so that it doesn't come across as a disjointed list of sound bites. — Carl (CBM · talk) 20:10, 13 February 2011 (UTC)

Mächtigkeit

I just want to clear up that in a mathematical context, "Mächtigkeit" means cardinality in German, not power. When applied referring to a set, this is its only and unambiguous meaning. To say in the article that it means power is not accurate nor helpful. It would be like saying that German articles about fields (Körper) are talking about "bodies". Or that German articles about sets (Mengen) are talking about "masses". Just as in English, German words have different meanings in a mathematical context, and it is exclusively in that context that they should be translated. Anything else is mistranslation, period.--75.80.43.80 (talk) 02:16, 19 March 2011 (UTC)

Cardinality is referred to as "power" in some (mostly older) English texts, so I don't think you can call it a flat mistake. It is no longer very usual to see this usage in English, though, so it should probably be limited to a historical context, and explained when it is used. --Trovatore (talk) 05:21, 19 March 2011 (UTC)

One example of this is Morley's paper "Categoricity in power" from 1965; the use of that particular phrase still seems to be common in model theory, but it's a vestigial usage. — Carl (CBM · talk) 13:07, 19 March 2011 (UTC)

Yes. I think in a historical discussion it can be appropriate to translate Mächtigkeit with power, since it is in part about the words used. In English one of the two terms died out, but in German both (Mächtigkeit and Kardinalität) continue to be used. Hans Adler 13:25, 19 March 2011 (UTC)

Removed Hume's principle section

I removed a section that was undeniably OR (as an IP editor pointed out). The section claimed that certain arguments and rebuttals were made, but it seems to be in the imagination of its author that these discussions take place. (The metaphor of computer-as-microscope makes clear that the argument is David Petry's, since it is his favorite personal metaphor.) If, indeed, there is evidence of this sort of argument in the literature, then let's find some citations rather than presenting our own version. Phiwum (talk) 13:34, 31 August 2011 (UTC)

"... found some acceptance ..." ?????

This is quite dismaying. I was referred to this page by an anti-Cantor crank using this article as support.

Cantor's theory is accepted by everyone as a theorem of Zermelo-Fraenkel set theory. You can use alternate set theories if you like, but no modern mathematician disputes the soundness of Cantor's diagonal argument.

For you to imply in the first para of this article that Cantor's diagonal argument is in some dispute among modern mathematicians, just gives ammo to the legion of online cranks.

This is a new low for Wiki in my opinion. Some responsible adult over there should fix this article to reflect modern mainstream mathematical thought; and not the ravings of the cranks.

What next, an article on angle trisection saying that "the proof that you can't trisect an angle with compass and straightedge is accepted by some, but I, Professor Crankenstein, have trisected the angle and published a pamphlet." I'm afraid to look.

76.102.69.21 (talk) 17:51, 11 May 2012 (UTC) steve@your-mailbox.com

The opening line certainly was misleading. I have edited it. Phiwum (talk) 19:33, 11 May 2012 (UTC)

Real numbers not bigger than Natural numbers?

The current article contains the claim:

"it follows that R has a different number than N. It does not prove, however, that the number of elements in R is in fact greater than the number of elements in N, for only the notion of two sets having different power has been specified; given two sets of different power, nothing so far has specified which of the two is greater."

However, as is mentioned in the article on Cardinality (http://en.wikipedia.org/wiki/Cardinality), there are three cases for comparing cardinality:

|A| = |B| : Exists a bijection from A to B

|A| ≥ |B| : Exists an injection from B to A

|A| > |B| : Exists an injection from B to A, but no bijection between the two

Creating an injection from N to R is trivial, it's the identity function. Therefore |R| > |N|.

Furthermore, this is consistent because there does not exist an injection f from R to N, simply because if such an f did exist, then it would be possible to invert it (call the inverse g), and if the domain of g were restricted to the set f(R), the co-domain would be R, and g would be a bijection from f(R), which is a subset of N, to R. We know that such a bijection cannot exist from the original diagonalization argument.

— Preceding unsigned comment added by 70.124.60.26 (talk • contribs)

I think that the author of that section is trying to sketch equality of cardinalities prior than introducing the < relation yet. If so, then it makes sense to say that "nothing so far has specified which of the two is greater," but I don't see the point of this presentation. The fact is that we can easily define the linear order, and it's a simple corollary from Cantor's Theorem that |R| > |N|. I'll change the text. Since this seems to be a discussion of a particular historical proof (Cantor's own proof), and I'm not familiar with that particular presentation, I'll leave the text as it is. I think the author is trying to mimic Cantor's own development of the theory. Phiwum (talk) 02:18, 12 September 2012 (UTC)

Removed "Other foundational controversies"

I removed the section "Other foundational controversies", reproduced below for convenience:

In 2012 and in a paper by J. E. Palomar Tarancón it is shown that Cantor's Theorem can be consequence of the existence of intrinsic set properties. Thus, Hume's principle need not be true, that is to say, if a bijection can be stated between two sets X and Y both must be equal in size. By contrast, although being equal in size, it is possible that no bijection can be stated between two sets. The proof has been published in Int. J. Open Problems Compt. Math., Vol. 5, No. 2, June 2012, in a paper entitled The Existence of Intrinsic Set Properties Implies Cantor's Theorem. The concept of Cardinal Revisited. (ref)

I did so because...

• The source seems questionable--lots of typos and many unclear statements. For instance, it talks about "size" all over but never defines it, even after saying, "It is worth mentioning that, although size and cardinality are frequently regarded as synonymous, in this paper cannot be identified."
• The journal seems superficially questionable; pretty new, open access, potentially lax publishing standards....
• The description above is very unclear. I think it's trying to say it might be the case that no "computable" bijection between a pair of sets exists, despite those sets having the same cardinality. If so, this strikes me as completely obvious and not at all worthy of mentioning here.

If someone finds worth in the article/paragraph, please feel free to rework it. 24.22.240.81 (talk) 05:18, 5 December 2012 (UTC)

I concur. The paper argues that size equivalence might not always correlate to bijection, so failing to define size is damning. His final corollary is that cantor's theorem can be derived from 2 parts of his previous 5-part theorem, except 1 of those parts he proved by showing a contradiction to cantor's theorem.

173.25.54.191 (talk) 03:24, 7 December 2012 (UTC)

---

It is written

As Leopold Kronecker claimed: "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there."

what is the reference to this quote??

edits by "renamed user 4"

Some of the material on this page was contributed by the banned user User:Renamed user 4. Thus, this edit from 2005 introduced a lot of questionable material, including some opinions attributed to R. Arthur. Any ideas whether this should be retained? Tkuvho (talk) 16:21, 16 December 2013 (UTC)

I found only 4 citations in Google Scholar of this article by Arthur dating from 2001. This does not justify inclusion, so I will deleted related material. Tkuvho (talk) 16:55, 17 December 2013 (UTC)

Inadequate statement

One paragraph reads:

"The assumption of the axiom of choice was later shown unnecessary by the Cantor-Bernstein-Schröder theorem, which makes use of the notion of injective functions from one set to another—a correlation which associates different elements of the former set with different elements of the latter set."

If one wants anyone to understand them, it is necessary to say what one means.

Here, I have to ask: The assumption of the axiom of choice was later shown unnecessary FOR WHAT ?????

I hope someone — someone who knows what that sentence is attempting to say — will fill in the blank in the article.Daqu (talk) 01:11, 6 December 2014 (UTC)

Please see the comments in the section Unmitigated nonsense above. Note in particular the comment that the article is sufficiently badly written that proposing it for deletion seems appropriate. Deletion seems appropriate to me, too. Lorenzo Traldi (talk) 09:38, 15 August 2015 (UTC)

Suggestion for improvement to "Cantors Argument"

I think the listed bullet points in this section would be much clearer and more well presented if the concept of cardinality were introduced and the precise term used to explain the argument instead of using number/size/power repeatedly. For example:

The set N of natural numbers exists (by the axiom of infinity), and so does the set R of all its subsets (by the power set axiom). By Cantor's theorem, R cannot be one-to-one correlated with N, and by Cantor's definition of number or "power", it follows that R has a different number than N. It does not prove, however, that the number of elements in R is in fact greater than the number of elements in N, for only the notion of two sets having different power has been specified; given two sets of different power, nothing so far has specified which of the two is greater.

vs.

The set N of natural numbers exists (by the axiom of infinity), and so does the set R of all its subsets (by the power set axiom). By Cantor's theorem, R cannot be one-to-one correlated with N, and it follows that the cardinality of R is different than that of N. It does not prove, however, that the cardinality of R is in fact greater than that of N, for only the notion of two sets having different cardinalities has been specified; given two sets of different cardinality, nothing so far has specified which of the two is greater. — Preceding unsigned comment added by 192.17.100.6 (talk) 14:46, 7 June 2016 (UTC)

Ross Finlayson and writing on usenet

Ross Finlayson is an author who has written prolifically to sci.math and sci.logic. He keeps the same argument claiming there is a unique counterexample to uncountability of the reals. He appears to elucidate on it quite regularly. — Preceding unsigned comment added by 75.172.120.163 (talk) 03:54, 11 June 2016 (UTC)

Finlayson is still quite at it on sci.math. He posted about ten posts today, and he never appears to post the same thing twice, while always he keeps repeating and building his notions of "A Theory" and this idea that "the continuum limit of the ratios of naturals as the denominator goes to infinity" is a function from the naturals whose range is [0,1]. Recently he posted a conversation with one of the new larger AI generators prompting it to agree with what he says, "extent, density, completeness, measure". He says that this "countable continuous domain" is non-contradictory with the uncountability of the complete ordered field because it's a result that thusly this continuum limit is "not a Cartesian function", and also that it falls out of all the uncountability arguments otherwise as un-contradicted, a counterexample in this way. Then he talks about Cantor space, 2^w, and some idea of "square Cantor space", and the AI generator arrived at some very suggestive names for these things, like "Dirichlet", "Cantor", ..., "Uniform Cantor function", "Dirichlet distribution staircase", "Cantor set indicator", "Harmonic Identity function", "Dirichlet distribution with unit mass", "Normalized antiderivative function", "Unit density", mathematically.

With the idea of _a_ function between the discrete and continuous, that's of course quite a regular notion, though, "not-a-real-function", that the idea of "three different continuous domains" now is usual, where "continuous domain" isn't even usually defined, usually. It's implicit. It's usually built up way higher, so having them down low non-contradictorily, and built standardly, modeled standardly as it were, as "infinite limits" or "continuum limits", not just these "line-reals" for "field-reals", then also some "signal-reals", three different continuous domains, it's something.

Also he defends all the standard things and demands their definitions stay the same.

It's involved. 97.113.23.37 (talk) 06:31, 1 March 2024 (UTC) — Preceding unsigned comment added by 97.126.122.83 (talk)

Look, the talk page is supposed to be used for discussing what should appear in the article, not just anything connected to the subject of the article. Are we going to discuss Finlayson in the article? Probably not, because I doubt there are reliable secondary sources on which to base such a discussion. So unless that changes, discussion of Finlayson's current activities here does not seem useful for figuring out what should appear in the article. --Trovatore (talk) 17:34, 7 March 2024 (UTC)

I read his material. To teach math he challenges his students with confusion math but believes Uncountable. Some others teach the same way. However he is wrong, confusing unimportant random reals that can be counted far down the set as important like the rationals and non randomly thus algorithmically produced irrationals. Victor Kosko (talk) 08:48, 12 April 2017 (UTC)

Clarification request of January 2014

Hi, RJGray, thank you for dealing with the clarification request. However, I still have a problem understanding the purpose of the involved sentence:

[The infinity axiom] allows us to prove, together with Cantor's theorem, that there exists at least one infinite set that cannot be correlated one-to-one with all its subsets.

Isn't this too obvious, the proof looking like this:

By the infinity axiom, there exists an infinite set N.
By Cantor's theorem, N, like every other set, cannot be correlated one to one with all its subsets.

Isn't it clearer to reorder the last two items as sketched below:

• By the infinity axiom, there exists an infinite set N.
• By the power set axiom, the set R of all subsets of N can be built.
• By Cantor's theorem, R cannot be one-to-one correlated with N; so R and N have different 'mightiness'.

- Jochen Burghardt (talk) 08:35, 7 October 2016 (UTC)

Hi Jochen, I'm glad you are interested in the improvement of this article. Thank you for your suggestions above. I am currently working on a rewrite of the "Cantor's argument" section and welcome your suggestions and the suggestions of others. It will probably be a few weeks before I have it done. I'll post my rewrite on one of my user pages so you and others may make suggestions or edit it. It turns out that in 1895 Cantor defined when one set is of less cardinality than another. By using this definition, there is a short proof that the cardinality of set A is less than the cardinality of P(A), the power set of A. Here's his definition: The cardinality of set A is less than the cardinality of set B if these sets satisfy the conditions: (1) there is an injective function from A to B, and (2) there is no injective function from B to A. - RJGray (talk) 14:27, 7 October 2016 (UTC)

Rewrite of "Cantor's argument"; will submit next week

I've incorporated Jochen Burghardt's suggestions in my rewrite of the "Cantor's argument" section, which is available at User:RJGray/Cantor's argument. I welcome comments and edits. I plan to submit it next week. I also rewrote the lead so it mentions material in all the sections.

I followed the original design of this section, which I like a lot. I like the combination of history with a modern version of Cantor's argument. I've expanded the history and added proofs to the argument. The new section mentions Cantor's 1874 proof because the old section said that Cantor identified the larger set that he defined in the 1891 article with the set of real numbers. The new section explains why he avoided the real numbers in his 1891 article.

I reordered the bulleted items in the argument to handle Jochen's suggestion while using just 4 items for this part of the argument. It now reads like a play that starts by introducing the cast of characters, in this case N and P(N). One of my favorite parts of the old section is the introduction of definition of "having the same number" that tells how this intuitive concept is captured by a set theoretic definition. This item is followed by an item using this definition to state a theorem. I repeat this introduction of a definition followed by a theorem with the last two items.

The history was tricky to get right. When I read Cantor's 1891 proof, I was a bit confused by his reference to his 1878 definition of "less than/greater than" cardinality. In 1878, he was dealing with two cardinalities, that of the set of natural numbers and the set of real numbers, so his definition worked for this case. I found it surprising that he would reference it as late as 1891. I mention his 1883 principle that explains why his definition worked in 1891.

The new section ends by following the design of the old section—namely, by finishing the argument using weaker assumptions than the axiom of choice. Instead of the Schröder–Bernstein theorem, I use Cantor's 1895 definition of "greater than" because it's his definition and it's the simplest way to complete the argument. Also, readers get a complete argument and don't have to look up the proof of the Schröder–Bernstein theorem to see how much set theory is needed to prove Cantor's theorem. The modern proof I give is not original. It's essentially Zermelo's 1908 proof except he only used Cantor's "greater than" definition and didn't bother giving a separate proof that P(A) and A have different cardinalities, which I included because I was following Cantor's argument.

I added a reference to the John P. Mayberry quote. I deleted "Few have seriously questioned this step" because I could not find it in Mayberry's book. If someone wants to put it back in with a reference that's fine with me. It's the type of statement that needs a reference. As far as terms go, the English translation of Cantor's 1895 definition uses the terms "aggregate" and "part" instead of "set" and "subset". So instead of using "power" that is a bit archaic like "aggregate" and "part", I prefer to use "cardinality", which also has the advantage that it can be linked to. I avoided "bijection" and "injection" because these are fairly recent terms introduced by Bourbaki. Cantor's German is closer to and is usually translated as "one-to-one correspondence". Of course, if I had used the Schröder–Bernstein theorem, I might have used Bourbaki's terms since they are used in the Schröder–Bernstein article. --RJGray (talk) 18:39, 3 November 2016 (UTC)

I have posted the rewrite. --RJGray (talk) 20:10, 10 November 2016 (UTC)

Equinumerous vs. equipollent vs. equipotent

Equinumerous vs. equipollent vs. equipotent

I've done searches in Wikipedia for "equinumerous set", "equipollent set", and "equipotent set" and found "equinumerous" is most common. I use "set" in the search because just the first word brings up the Equinumerosity article. I also did searches on Google books for "equinumerous" and found it and "equipotent" about equally common, though "equinumerous" appears more in popular books. I also looked into "equinumerous with" vs. "equinumerous to" and found the first to be slightly more popular. So I've changed "equipollent to" to "equinumerous with". I also think this change fits in well with the 3rd step of the argument that now reads: "The concept of 'having the same number' … Sets in such a correlation are called equinumerous …" --RJGray (talk) 20:07, 14 November 2016 (UTC)

To my ear, "equipollent" and "equipotent" sound a bit dated. Of course this article is (largely) about a historical controversy, but that doesn't mean we can't use modern language. --Trovatore (talk) 20:48, 14 November 2016 (UTC)

Thanks for your feedback, I agree with both of your points. --RJGray (talk) 21:35, 18 November 2016 (UTC)

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The Infinite-Set Inconsistency

Not quite sure how or if to add this as I think my edits were correctly rejected by Wikipedia policies as own-work (I am new to editing). My concern is that "Controversy over Cantor's theorem" spends most of its time trying to justify Cantor by re-presenting the power-set based proof / diagonal argument. It strikes me this page is specifically about showing the evidence against the mainstream viewpoint.

It is wrong to state those who disagree with Cantor are either constructionist or finitist.

Both "countable-infinty" and "infinite-set" are inherently inconsistent mathematical concepts. This does not relate to self-evidence or finitism but to self-consistency and the essential distinction between unbounded and infinite within algebraic mathematics. Within analysis we may consider the limit, so effectively allow "finite" to become "infinite" but within algebra this is mathematically illegal. See and . Epdarnell (talk) 12:26, 26 August 2019 (UTC)

A more balanced discussion of the finite/infinite alternative, with some references, is available here: v:WikiJournal Preprints/Can each number be specified by a finite text?#Finite set theory. Boris Tsirelson (talk) 12:49, 26 August 2019 (UTC)

@Epdarnell: It's not just that you're using your own work, but that your work isn't published in a reliable source – it's published on a site that allows anyone to post anything. You can't use it here for the same reason that I can't write a paper that the Moon is made out of cheese, post it, and then insist that it be cited in our article about the Moon as an alternative viewpoint about lunar composition. –Deacon Vorbis (carbon • videos) 13:12, 26 August 2019 (UTC)

@Epdarnell: Fixing ping –Deacon Vorbis (carbon • videos) 13:13, 26 August 2019 (UTC)

References

J A Perez: (un)due weight?

About current edit by 83.248.229.13 reverted by Jochen Burghardt: I did not find any relevant article that cites "Addressing mathematical inconsistency: Cantor and Godel refuted" by J.A. Perez (what a sexy title, by the way, wow...). Thus I wonder, is it worth to be mentioned here? Boris Tsirelson (talk) 14:06, 20 November 2019 (UTC)

The statement in the article: "Others have also taken issue with Cantor's proof regarding the cardinality of the power set.[15][16]" is sourced to two papers, neither of which are published or by credentialed mathematicians. As such, this borders on WP:OR, and OR with poor primary sources to boot. It should simply be removed altogether. –Deacon Vorbis (carbon • videos) 14:48, 20 November 2019 (UTC)

I agree. Paul August ☎ 16:43, 20 November 2019 (UTC)

My motivation for my revert was that the reference shouldn't be simply deleted by an anonymous IP referring to a dubious blog entry for justification.

After glancing at the Perez and the Zenkin paper, I agree that none of them appears to be mathematically sound.

Interestingly, both seem to argue that the one set proven not to be in the range of the mapping could easily be added (Zenkin: abstract, line 3; theorem 1, line -5; Perez switches to binary strings on p.10, arguing that only one disagreement string exists in par.2, line 3). Maybe, this common flaw should be commented upon in the article (I didn't check if this is already present).

Possibly both references could serve as examples for flawed proofs. If this doesn't match wikipedia policy, I agree to delete them. - Jochen Burghardt (talk) 17:11, 20 November 2019 (UTC)

Historically this article has been largely about crank or crackpot attempted disproofs of Cantor rather than the more serious philosophical arguments, so "not mathematically sound" is not in itself a reason to remove them. But I do think we should stick to the more "notable" ones. Zenkin I have heard of; Perez I have not. That doesn't prove anything in itself, but my guess is that this difference will show up in the number of RSS that bother to cover the two individuals. --Trovatore (talk) 01:57, 21 November 2019 (UTC)

Non neutral point of view

I'll tag the article with {{Unbalanced}} for the following reasons:

• A non-expert reader that reads this article cannot imagine that all modern mathematics is built on Cantor's views that sustain Zermelo–Fraenkel set theory. Even the theories that reject infinite sets use Zermelo–Fraenkel theory for proving their relative consistency. The fact that modern mathematics is based on Cantor's views appears only in two obscure sentences in the lead, and is not developped in the article.
• This is true that, at the end of the 19th century, and at the beginning of the 20th century there was a controversy, but only the mathematicians that rejected Cantor's views are cited. There is no quotation of the numerous mathematicians that supported Cantor's theory, and no mention of those who not only supported Cantor's view, but have also worked for fixing apparent inconsistencies (Russel's paradox).

D.Lazard (talk) 15:49, 16 June 2020 (UTC)

Heh heh heh you should see what it used to be like. This article has a fraught history. It originated as a sympathetic account of the views of some net.cranks who hung around USEnet. It's been much cleaned up since then, but I don't think the article ever quite decided exactly what it wanted to be about. Both the contemporary controversy around Cantor, and modern schools that reject the completed infinite (or other concepts used by Cantor), are notable topics, but the idea of a smooth transition from the article's starting point to a good article about those topics never sounded like a very sound plan to me. --Trovatore (talk) 18:20, 16 June 2020 (UTC)

WP:TNT is possibly the best solution. I am not willing to rewrite this article. This is the reason of my tag, aimed to warn non-experts about the problem. D.Lazard (talk) 19:25, 16 June 2020 (UTC)

I'd agree to delete this article. The description of Cantor's theorem may be found in about a dozen other articles. Some of the historical citations might be moved to appropriate articles about intuitionism, constructive set theory, etc., to garnish them with some philosophical viewpoints. - Jochen Burghardt (talk) 15:05, 17 June 2020 (UTC)

Covering arguments in unreliable sources

Jochen undid a delete of sources by Tercer, which I have in turn undone.

The text I redeleted was Others have also taken issue with Cantor's proof regarding the cardinality of the power set. supported by two notes: Perez, Juan A. (2010). "Addressing mathematical inconsistency: Cantor and Gödel refuted". arXiv:1002.4433 [math.GM]. and Zenkin, Alexander. "Cantor's Diagonal Argument: A New Aspect" (PDF). Dorodnitsyn Computing Center. Retrieved 2 October 2014..

I don't think we should link to these articles without content that competently deals with the issues raised by them. There is a WP:UNDUE issue here of giving fringe viewpoints the last word. Nonetheless, Jochen is right that we should cover the content of these views. — Charles Stewart (talk) 11:43, 12 May 2021 (UTC)

@Chalst and Tercer: I completely agree with you. The articles (or similar ones at the border to fringe math) should be mentioned, but with appropriate comment. I think presenting dubious viewpoints (and hence with unreliable sources) is unavoidable here, to demonstrate the full range of viewpoints about Cantor's theory. I'd like to ask RJGray, the main contributor, to suggest an appropriate treatment. - Jochen Burghardt (talk) 12:16, 12 May 2021 (UTC)

I disagree. WP:UNDUE is explicitly that we shouldn't cover fringe views: Generally, the views of tiny minorities should not be included at all, except perhaps in a "see also" to an article about those specific views. Also, WP:SPS is clear that we should not use self-published sources unless they are written by recognized experts. This is not the case here. These sources are simply mathematically mistaken. It is not a matter of opinion. They have no place in Wikipedia. Tercer (talk) 12:19, 12 May 2021 (UTC)

I'm not an expert in this area of history of mathematics (, but RJGray is). Moreover, I'm not familiar with all subtleties of WP:FRINGE. That said, my impression is that unusually many (near-)fringe publications about Cantor's theory are around (like about squaring the circle and doubling the cube), so tiny minorities wouldn't apply. And I feel Wikipedia policy does allow to mention it when many fringe views exist with respect to an article; this is what e.g. Squaring_the_circle#Other_modern_claims does (it is one of the exceptions admitted by generally, I guess). Provided (near-)fringe approaches are similarly popular here as in the circle article (which I can't really judge), mentioning this, along with an example, should be OK, or shouldn't it? - Jochen Burghardt (talk) 13:02, 12 May 2021 (UTC)

The article is "Controversy over Cantor's theory", and mostly covers serious philosophical arguments from respected mathematicians. It is not "Fringe views on Cantor's theory", where it might be appropriate to cite unreliable sources simply to remark on their existence. Even then, we should mention noteworthy fringe views, not some random mistakes made by random people. The article by Perez is unpublished and has been cited 0 times, and the one by Zenkin is not even on the arXiv, probably has also never been cited. Tercer (talk) 14:06, 12 May 2021 (UTC)

Zenkin's article, along with many of his other writings on Cantor's theorem, was cited in an apparently unpublished article by Alexey Stakhov, who I understand is a serious historian of mathematics, in http://www.trinitas.ru/rus/doc/0016/001g/3965-sth.pdf. It was Stakhov's article that made me take a closer look at Zenkin's article. Our rules on reliable sources are not a litmus test of fringeness. — Charles Stewart (talk) 14:35, 12 May 2021 (UTC)

From the article: By the way, in the Wikipedia article [5] there are the links to two Zenkin’s critical articles [7,8], concerning Cantor’s theory of infinite sets. I find it hard to take this reference seriously. Tercer (talk) 14:57, 12 May 2021 (UTC)

Indeed. Moreover, the existence of a large number of fringe-y writings on a topic doesn't necessarily mean that there is a singular, non-mainstream viewpoint that has a substantial following. Rather, there can be many different fringe ideas, each one held by a tiny minority. In a case like Squaring the circle#Other modern claims, we have secondary sources indicating the significance of each claim. Most pseudomathematics doesn't reach that status. XOR'easter (talk) 16:33, 12 May 2021 (UTC)

• I've taken a closer look at the Zenkin paper. Zenkin earlier published Logic of actual infinity and G. Cantor's diagonal proof of the uncountability of the continuum, Review of Modern Logic (a now defunct but respectable open access logic journal), 9(3-4):27-82, 2004 - the article essentially takes Wittgenstein's mathematics-as-activity idea suggested in Hodges' paper and develops it at length into an argument that Cantor's proof proves uncountability of the continuum, but in a way that undermines existence claims for uncountable sets. This is not fringe. The self-published article is strange, but I think because it is poorly written, not because the ideas are beyond the pale. I support excluding the deleted articles, if only because this is one of the worse articles to open to self-published works, but I do think we should be broadminded in documenting the range of views on the topic. — Charles Stewart (talk) 13:51, 12 May 2021 (UTC)

Ross Finlayson's developments about set theory and real numbers

Ross Finlayson defines continuous domains and definitions of continuity beyond Dedekind's. It results a sort of controversy that needs to be resolved in the set theory. He defines at least three continuous domains and the first one is like [0,1] split into infinitesimals, that he calls "line-reals" then second is the complete ordered field, "field-reals", then a third is about double-density and Nyquist/Shannon the "field reals". It really speaks to the continuum being elementary. Having more than one definition of continuity, is more than the standard has. He made his line-reals all formal with least upper bound and sigma algebras, then says that it doesn't contradict Cantor while still being countable. I like it a lot. 97.126.122.83 (talk) 16:49, 7 March 2024 (UTC)