Obections to Cantor's Theory (Wikipedia article)
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From: Rick Decker on 19 Jul 2005 20:16 David Kastrup wrote: > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > >>David Kastrup said: >> >>>Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >>> >>> >>>>David Kastrup said: >>>> >>>>>Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >>>>> <mega-snip> > > Of course, this is complete and utter bullshit, since N is not a > natural number to start with, but even if we dive into your utter > bullshit with a vengeance, it does not hold water. > David, while I admire your Quixotic attempt to ram some sense into the heads of the terminally ineducable, you owe me the price of a new Mixed Metaphor Detector (TM), since you just overloaded mine. <snip> Regards, Rick
From: Dik T. Winter on 19 Jul 2005 21:20 In article <85zmsic1xg.fsf(a)lola.goethe.zz> David Kastrup <dak(a)gnu.org> writes: .... > 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. Well, you know, some people react strange. I have had a similar argument with two persons in the last few months. Both agreed that each step was correct, and both said also that the proof was nevertheless wrong. In one case the reason was that taking the power-set of the naturals was invalid to begin with. The other is still talking about the impossible set. Hrm. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 19 Jul 2005 21:36 In article <MPG.1d47123542f108d6989f25(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Even though every subset of the natural numbers can be represented by > a binary number where the first bit denotes membership of the first > element, the second bit denotes membership of the second element, > etc? The only objection to this bijection between the natural numbers > and the subsets of the natural numbers is the nonsensical insistence > that every natural number in the infinite set is finite, which is > mathematically impossible, given the fact that each additional > element requires a constant incrementation of the entire range of > values in the set. TO climbs on his hobby horse again and gallops off in all directions. Every non-empty set of naturals has a smallest member. The set of infinite naturals is, and will remain, empty until some smallest infinite natural can be shown to result from adding 1 to some (largest) finite natural.
From: Virgil on 19 Jul 2005 21:43 In article <MPG.1d4712ec2f75d957989f26(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Jesse F. Hughes said: > > 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? > > > > > In all actuality, the flaws in various proofs and assumptions in set theory > have been directly addressed, and ignored by the mainstream thinkers here. TO has been using a faulty address book, then, as he has found or addressed any actual flaw. > > Now, I am not familiar, I think, with the proof concerning subsets of the > natural numbers. Certainly a power set is a larger set than the set it's > derived from, but that is no proof that it cannot be enumerated. The proof that there is no surjection f:S -> P(S) from any set to its power set is established by showing that for any such f, there are sets not of form f(x), for any x in S. > Is this the > same as the proof concerning the "uncountability" of the reals? Quite similar, but not identical. The basis is the same: showing that any mapping from the smaller to the larger _must_ fail to be surjective.
From: Virgil on 19 Jul 2005 21:51
In article <MPG.1d4715811811605b989f27(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > David Kastrup said: > > 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. > > > > > Is the above your 7-line proof? it makes no sense. It makes sense to those who have sufficient mental capacity to understand it. Those who either cannot or will not understand it, but cannot fault it, are irrelevant. There is no reason to > expect > the natural number corresponding to the subset to be a member of that subset. But it transpires that no natural can get mapped to this well-defined set, so there is a necessary failure to surject. > if this rests on the diagonal proof, there is a very clear flaw in that proof > which you folks simply dismiss as irrelvant, but which is fatal. Still, > discussing these things with Cantorians is like trying to discuss evolution > with an evangelical christian. Only if TO is taking the part of the evangelicals, making claims without proofs, and, like the evangelicals, ignoring the opposition's proofs. |