Obections to Cantor's Theory (Wikipedia article)
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From: Jean-Claude Arbaut on 19 Jul 2005 10:10 Le 19/07/05 1:02, dans 1121727755.158001.288300(a)g44g2000cwa.googlegroups.com, ýýdavid petryýý <david_lawrence_petry(a)yahoo.com> a ýcritý: > > 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. > > *** > [...] Some anti-Cantorian try to pretend that set theory is logically wrong. We had recently (yesterday and the day before) a discussion with a french anti-cantorian on fr.sci.maths, who is also a contributor to Wikipedia. Sadly, he proved to have a very poor knowledge of set theory and logic (or was dishonest enough not to answer seriously - read, without insults and repetition of the same absurdities - any argument). I hope Wikipedia won't become a stronghold for bogus mathematics... As another poster noticed in this thread, it is perfectly understandable to discuss "physical existence" of sets or anything mathematical. It is also understandable to suggest other ways, but not to slander a theory by wrong arguments. Good luck.
From: Jean-Claude Arbaut on 19 Jul 2005 10:16 Le 19/07/05 15:59, dans mckenzie-60C274.14595219072005(a)news.aaisp.net.uk, ýýAlec McKenzieýý <mckenzie(a)despammed.com> a ýcritý: > 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. There is no certainty. Set theory has not been proved to be coherent. But all incoherencies so far have been solved or dismissed. That's the way mathematics work. If you can read french, here is a very interesting article on that subject: http://1libertaire.free.fr/godel02.html
From: Jean-Claude Arbaut on 19 Jul 2005 10:20 Le 19/07/05 16:01, dans 1121781687.298529.132000(a)g44g2000cwa.googlegroups.com, ýýRoss A. Finlaysonýý <raf(a)tiki-lounge.com> a ýcritý: > Obviously I suggest the null axiom theory. ZF is inconsistent. As a joke, it's funny... But if you have a real proof, instead of writting promises, write your proof.
From: David Kastrup on 19 Jul 2005 10:20 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. 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. Just what _would_ constitute justification in your book? -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Jesse F. Hughes on 19 Jul 2005 10:27
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 is easily formalized. It is a remarkably short and simple proof and does not require any large body of theorems to reach its conclusion. (It does require unpacking of the definitions of "function" and "onto", of course, and this is tedious but not difficult.) I don't know what version of the proof David has in mind, but I'm sure that a formal version is longer than seven lines. Nonetheless, precisely because it is formal, that version can be easily checked for correctness. >> 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. What is your standard of justification? What more should a person do than provide the proof? Seems like you want a proof that the proof is correct in addition to the proof itself. Of course, we would want further proof that this verification is correct. And then... In any case, none of this applies much to Cantor's remarkably simple argument. Just work through it in first order logic. Nothing to it! Are you waiting for someone else to do this work for you? Well, no problem. I understand it's been done in Isabelle, for instance. See <http://arxiv.org/pdf/cs.LO/9311103>. But I'm also sure that there are other formalizations lying around. -- Jesse F. Hughes "We need to counter the shockwave of the evildoer by having individual rate cuts accelerated and by thinking about tax rebates." -- George W. Bush, Oct. 4, 2001 |