Obections to Cantor's Theory (Wikipedia article)
in [Logic]
|
Prev: Derivations
Next: Simple yet Profound Metatheorem
From: Jesse F. Hughes on 19 Jul 2005 03:41 quasi <quasi(a)null.set> writes: > On 18 Jul 2005 16:02:35 -0700, "david petry" > <david_lawrence_petry(a)yahoo.com> 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. >> >> .... >> >>These "anti-Cantorians" see an underlying reality to >>mathematics, namely, computation. They tend to accept the >>idea that the computer can be thought of as a microscope >>into the world of computation, and mathematics is the >>science which studies the phenomena observed through that >>microscope. They claim that that paradigm includes all >>of the mathematics which has the potential to be applied to >>the task of understanding phenomena in the real world (e.g. >>in science and engineering). > > To me, it sounds very presumptuous to talk about anti-Cantorians as if > they were a well defined group. Much of what you say strikes me as > your own opinion. essentially an editorial, but camouflaged by > attributing the views to a group. If you could even assemble a group > of anti-Cantorians -- try it, I dare you -- I'll bet they would > disagree with each other on almost everything. Exactly. The quoted paragraph gives it away. This isn't about "anti-Cantorians", whatever the hell that might mean. This is about Petrians, a well-defined group in which there is no dissension at all (since there is only one member). Well-put. -- "I am one of the more important discoverers in mathematical history, but future students will have the luxury of knowing that, and may be puzzled by your behavior now." -- James Harris (At least I have the foresight to quote his pearls of wisdom.)
From: Alec McKenzie on 19 Jul 2005 04:34 "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. Yet that is just what pro-Cantorians appear to be asserting, with no justification that I can see. -- Alec McKenzie mckenzie(a)despammed.com
From: Barb Knox on 19 Jul 2005 04:55 In article <mckenzie-9FA4AC.09344519072005(a)news.aaisp.net.uk>, Alec McKenzie <mckenzie(a)despammed.com> wrote: > "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. Actually, in this case it does. The proof is simple enough to be thoroughly check by humans and computers. There is no flaw in the proof. >Yet >that is just what pro-Cantorians appear to be asserting, with no >justification that I can see. Do you dispute the above justification? -- --------------------------- | BBB b \ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | -----------------------------
From: David Kastrup on 19 Jul 2005 04:56 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. 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. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Jesse F. Hughes on 19 Jul 2005 04:51
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. Yet > that is just what pro-Cantorians appear to be asserting, with no > justification that I can see. Huh? The proof of Cantor's theorem is easily formalized. It's remarkably short and simple and every step can be verified as correct. It is perfectly reasonable to assert that no such flaw exists (given the axioms used in the proof). Indeed, why would anyone entertain any doubts when he can confirm the correctness of each and every step of the proof? -- Jesse F. Hughes "How come there's still apes running around loose and there are humans? Why did some of them decide to evolve and some did not? Did they choose to stay as a monkey or what?" -Kans. Board of Ed member |