Talk:Controversy over Cantor's theory
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- 1 Cantor Anti-Diagonal Argument — Clarifying Determinateness and Consistency in Knowledgeful Mathematical Discourse
- 2 A Few Computer Scientists Know Think Cantor Diagonal is Wrong
- 3 Removal of Bishop quote
- 4 Time frame; synthesis
- 5 Zenkin Gets No Mention
- 6 Zenkin
- 7 Unmitigated nonsense
- 8 Cohen has closed the controversy by own proof
- 9 lede tone
- 10 Mächtigkeit
- 11 Removed Hume's principle section
- 12 "... found some acceptance ..." ?????
- 13 Real numbers not bigger than Natural numbers?
- 14 Removed "Other foundational controversies"
- 15 edits by "renamed user 4"
- 16 Inadequate statement
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.
- 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)
- You are absolutely correct. I would be classified a constructionist, I suppose.WaywardGeek (talk) 22:07, 2 June 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)
- 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)
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.126.96.36.199 (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. 188.8.131.52 (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.
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:
- 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."
- 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 184.108.40.206 (talk) 02:00, 26 March 2010 (UTC)
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. 220.127.116.11 (talk) 22:29, 29 March 2010 (UTC)
- Oh dear, this bullshit was added by Peter Damian in November 2005!  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 18.104.22.168 (talk) 04:32, 11 September 2010 (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.
- 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)
- "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)
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.--22.214.171.124 (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)
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.
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.
- 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.
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.
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)
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 ?????