Obections to Cantor's Theory (Wikipedia article)
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From: Jean-Claude Arbaut on 19 Jul 2005 10:57 Le 19/07/05 16:20, dans 85zmsic1xg.fsf(a)lola.goethe.zz, ýýDavid Kastrupýý <dak(a)gnu.org> a ýcritý: > Alec McKenzie <mckenzie(a)despammed.com> writes: > >> David Kastrup <dak(a)gnu.org> wrote: >> >>> Alec McKenzie <mckenzie(a)despammed.com> writes: >>>> It has been known for a proof to be put forward, and fully accepted >>>> by the mathematical community, with a fatal flaw only spotted years >>>> later. >>> >>> In a concise 7 line proof? Bloody likely. >> >> I doubt it had seven lines, but I really don't know how many. >> Probably many more than seven. > > It was seven lines in my posting. You probably skipped over it. It > is a really simple and concise proof. Here it is again, for the > reading impaired, this time with a bit less text: > > Assume a complete mapping n->S(n) where S(n) is supposed to cover all > subsets of N. Now consider the set P={k| k not in S(k)}. Clearly, > for every n only one of S(n) or P contains n as an element, and so P > is different from all S(n), proving the assumption wrong. Not sure what you call a complete mapping. If it's a surjective application, then you only proved that you cannot find one. Big deal. Hint: it has been known since the early days of set theory. > So now it is 4 lines. And one does not need more than that. > >>> And that's what you call "with no justification that I can see". >> >> No, it is not -- what I called "with no justification that I can >> see" was something else: >> >> It was the assertion that no flaw having been found in a proof leads >> to a certainty that such a flaw cannot exist. I still see no >> justification for that. > > Fine, so you think that a four-liner that has been out and tested for > hundreds of years by thousands of competent mathematicians provides no > justification for some statement. Depends on which statement. If you want to prove that set theory is not coherent, you failed.
From: Patrick on 19 Jul 2005 11:11 Alec McKenzie wrote: > David Kastrup <dak(a)gnu.org> wrote: > > >>Alec McKenzie <mckenzie(a)despammed.com> writes: >> >> >>> "Stephen J. Herschkorn" <sjherschko(a)netscape.net> wrote: >>> >>> >>>>Can anti-Cantorians identify correctly a flaw in the proof that >>>>there exists no enumeration of the subsets of the natural numbers? >>> >>>In my view the answer to that question a definite "No, they can't". >>> >>>However, the fact that no flaw has yet been correctly identified >>>does not lead to a certainty that such a flaw cannot exist. >> >>Uh, what? There is nothing fuzzy about the proof. > > > I am not suggesting there is anything fuzzy about the proof. > > >>Suppose that a mapping of naturals to the subsets of naturals exists. >>Then consider the set of all naturals that are not member of the >>subset which they map to. >> >>The membership of each natural can be clearly established from the >>mapping, and it is clearly different from the membership of the >>mapping indicated by the natural. So the assumption of a complete >>mapping was invalid. >> >> >>>Yet that is just what pro-Cantorians appear to be asserting, with no >>>justification that I can see. >> >>Uh, where is there any room for doubt? What more justification do you >>need apart from a clear 7-line proof? It simply does not get better >>than that. > > > I quite agree that it does not get better than that, but I think > one must allow some room for doubt, however small, for any > proof. Otherwise one is proclaiming infallibility. > > It has been known for a proof to be put forward, and fully > accepted by the mathematical community, with a fatal flaw only > spotted years later. Not a 5-liner.
From: David Kastrup on 19 Jul 2005 11:41 Jean-Claude Arbaut <jean-claude.arbaut(a)laposte.net> writes: > Le 19/07/05 16:20, dans 85zmsic1xg.fsf(a)lola.goethe.zz, ýýDavid Kastrupýý > <dak(a)gnu.org> a ýcritý: > >> Alec McKenzie <mckenzie(a)despammed.com> writes: >> >>> David Kastrup <dak(a)gnu.org> wrote: >>> >>>> Alec McKenzie <mckenzie(a)despammed.com> writes: >>>>> It has been known for a proof to be put forward, and fully accepted >>>>> by the mathematical community, with a fatal flaw only spotted years >>>>> later. >>>> >>>> In a concise 7 line proof? Bloody likely. >>> >>> I doubt it had seven lines, but I really don't know how many. >>> Probably many more than seven. >> >> It was seven lines in my posting. You probably skipped over it. It >> is a really simple and concise proof. Here it is again, for the >> reading impaired, this time with a bit less text: >> >> Assume a complete mapping n->S(n) where S(n) is supposed to cover all >> subsets of N. Now consider the set P={k| k not in S(k)}. Clearly, >> for every n only one of S(n) or P contains n as an element, and so P >> is different from all S(n), proving the assumption wrong. > > Not sure what you call a complete mapping. If it's a surjective > application, then you only proved that you cannot find one. Big > deal. Hint: it has been known since the early days of set theory. Well, tell that to Alec. >> So now it is 4 lines. And one does not need more than that. >> >>>> And that's what you call "with no justification that I can see". >>> >>> No, it is not -- what I called "with no justification that I can >>> see" was something else: >>> >>> It was the assertion that no flaw having been found in a proof leads >>> to a certainty that such a flaw cannot exist. I still see no >>> justification for that. >> >> Fine, so you think that a four-liner that has been out and tested for >> hundreds of years by thousands of competent mathematicians provides no >> justification for some statement. > > Depends on which statement. If you want to prove that set theory is > not coherent, you failed. Well, tell that to Alec. Perhaps you need to reread the thread and find out who is writing what? -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Kevin Delaney on 19 Jul 2005 11:57 david petry wrote: > I'm in the process of writing an article about > objections to Cantor's Theory, which I plan to contribute > to the Wikipedia. I would be interested in having > some intelligent feedback. Here' the article so far. > It seems to me that gap between "Cantorians" and "Anti-Cantorians" is pretty much the same gap between classical and modern thought as existed in most subjects in the 20th century. As I recall, Wallace described transfinite theory as the final nail in Aristotle's coffin. Transfinite theory was the primary reason for removing the study of logic, grammar and arithematic in American public education and replacing it with new math. There are some who think the world was on a better thread with the classical tradition. > The pure mathematicians tend to view mathematics as an art > form. They seek to create beautiful theories, Many of the anti-Cantorians are such because they find the theory to be ugly. Poincare described it as a disease. Brouwer had similar invective. I think it isn't quite accurate to say that the people who dislike the theory are completely devoid of aesthetics. A better description is that they are petty bourgeoisie. Cantorians, of course, are avante garde revolutionaries. As pointed out in a different post. The people who are opposed to a theory are generally a lot more diverse than those who support it. For example, Brouwer was opposed to the law of excluded middle. I suspect that others are hoping to pull off a stunt like Russell. Russell's early work on the reflexive paradox could be considered "anti-Cantorian". Russell's two step catapulted him to the top of the intellectual community and led to the decline of Frege. I suspect that many people toy with anti-Cantorian thoughts because they hope to repeat the trick.
From: Alec McKenzie on 19 Jul 2005 12:19
David C. Ullrich <ullrich(a)math.okstate.edu> wrote: > Ok, here's another question. Suppose that we want to > prove that A implies B. Suppose that we have an > completely flawless proof that A implies C, and > a completely flawless proof that C implies B. > Does the union of those two proofs constitute > a flawless proof that A implies B? Yes, I would say it does. > I imagine you'll say yes to that as well. But > the proof of the theorem in question really > does involve nothing more than statements > which are true by definition and statements > which follow from previous statements by > "if A implies C and C implies B then A implies > B" arguments. I would expect that to be the case for most direct proofs, if not all. In the case of the proof of the theorem in question, we do not already know for a fact that the conclusion is true; neither do we know that it is false. If we did already know it was true there would be little point in trying to find a flaw, regardless of whether one might exist (there could be a flaw in the proof, even if the conclusion is correct). But if we knew it to be false, the flaw would have to be there even if we cannot find it. There are also proofs where we do know for a fact the conclusion is false, even though there is no apparent flaw. The paradox of the unexpected examination is an example of this, and I think that if the conclusion in that case had been one, like Cantor, where we had no way of knowing (apart from the proof) whether the conclusion were true or false, the validity of the proof would be almost universally accepted, and just as vigorously defended. Equally, if the conclusion of Cantor's proof were known for a fact to be false, it would be known as Cantor's paradox. My own feeling is that there exists the possibility, however slight, that Cantor's conclusion is an obscure manifestation of a paradox. -- Alec McKenzie mckenzie(a)despammed.com |