Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 20 Jul 2005 10:59 stephen(a)nomail.com said: > In sci.math Barb Knox <see(a)sig.below> wrote: > > In article <MPG.1d4726e11766660c989f2f(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > [snip] > > >>Infinite whole numbers are required for an infinite set of whole numbers. > > > Good grief -- shake the anti-Cantorian tree a little and out drops a > > Phillite. Here's a clue: ALL whole numbers are finite. Here's a > > (2nd-order) proof outline, using mathematical induction (which I > > assume/hope you accept): > > 0 is finite. > > If k is finite then k+1 is finite. > > Therefore all natural numbers are finite. > > Talking to Tony is a waste of time. He does not understand > induction and is a firm believer in "after infinity". He > is a fine example of the non-mathematical sort who complains > about Cantor. > > Stephen > Nice ad hominem. You never understood any of my points. Take your fingers out of your ears and stop with the "blah blah" -- Smiles, Tony
From: Tony Orlow on 20 Jul 2005 11:09 Virgil said: > 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. > And why is a lrger set necessarily uncountable, or unenumerable? If you can draw bijections between naturals and evens and declare them the same, when one is obviously twice the size of the other, then why can't a similar bijection be created. After all, wouldn't you say that the set of all integral powers of 2 is a countable set? Or all log2's of natural numbers? In my book, there are 2^N log2's of natural numbers. That doesn't make it uncountable. It just makes it a bigger set. -- Smiles, Tony
From: Tony Orlow on 20 Jul 2005 11:12 Virgil said: > In article <MPG.1d471fc316a53825989f29(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > 100100...100100100100 Of course, you will argue that this infinite > > value is not a natural number, since all naturals are finite, but > > that is clearly incorrect, as it is impossible to have an infinite > > set of values all differing by a constant finite amount from their > > neighbors, and not have an overall infinite difference between some > > pair of them, indicating that at least one of them is infinite. > > TO's assumption of "infinite naturals" carefully avoids any > consideration of the concomitant necessity of at some point having to > add 1 to a finite number to produce an infinite number. > That is not only not a necessity, but an impossibility, as I have said many times, and with which you agree. The problem of the largest finite or smallest infinite cannot be solved, and should not be the centerpiece of a theory that hopes to achieve anything. -- Smiles, Tony
From: Tony Orlow on 20 Jul 2005 11:19 Dik T. Winter said: > In article <190720051211351753%suh(a)math.ucdavis.nospam.edu> Chan-Ho Suh <suh(a)math.ucdavis.nospam.edu> writes: > > In article <mckenzie-3D5A6E.14153919072005(a)news.aaisp.net.uk>, Alec > > McKenzie <mckenzie(a)despammed.com> wrote: > ... > > > 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. > > > > I doubt this. "Fully accepted" means that the community either was not > > paying attention or didn't care enough to check themselves, etc. I > > know of no proof in the modern literature that was verified correct by > > a large number of experts and with a flaw only found years later. > > What is modern in this sense? Many flawed proofs were (and are) based on > hidden assumptions, that everybody believed (believes) to be true. Euler's > proof of FLT for n=3 was based on such a hidden assumption. For many years > it was unchallenged, but later the hidden assumption was shown to be present > in the proof (and later proven to be indeed true, yes, that can happen, that > is why it is still call Euler's proof). Similar for Lamý's proof of FLT, > which had also a hidden assumption: Q(sqrt(p)) is an UFD. But in this case > that assumption was shown to be false (by Kummer). In both cases many of the > mathematicians of that time thought the proofs to be valid, but they were not > (in one case because there was a gap, in the other case because the assumption > was not valid). FLT is particularly interesting because it is fraught with > proofs that were considered to be valid at some time. (And indeed, Wiles' > first proof had an error, but that was detected before it was really > considered to be a proof. On the other hand, most FLT provers make similar > mistakes as have been made before again and again.) > > Now this case is quite dissimilar. Cantor's proof (about the cardinality > of powersets vs. the cardinality of the base sets) is indeed simple and > can be written in very few steps. If (as Alec McKenzie appears to think) > it might be a paradox, there should be a proof (based on the same axioms) > that is in contradiction with Cantor's proof. As Cantor's proof is valid > for finite sets, the contradiction proof should give the same result for > finite sets, but not for infinite sets. But the counter-proofs we see > either do not use the same axioms, do not use the same terminology, or > have a gap. > You know, if the only conclusion drawn from Cantor's proofs was that a power set is necessarily larger than its base set, I would have absolutely no problem. That fact is always the case and I don't dispute it. Infinite sets of finite natural numbers, on the other hand, are self contradictory, and the conflation of "larger and infinite" with "uncountable" is arbitrary. The diagonal proof has hidden assumptions that mathematicians seem to acknowledge and not question, and discussions of the relative infinites of digital numbers, tree parts, subsets etc, seem to purposely ignore the properties of the elements of which they are constructed. Being "anti-Cantorian" doesn't mean you disagree with everything he said, just with the stuff that contradicts other math and reality. -- Smiles, Tony
From: Tony Orlow on 20 Jul 2005 11:24
Dik T. Winter said: > In article <MPG.1d4722e516a9e4df989f2b(a)newsstand.cit.cornell.edu> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > ... > > The only reason to reject this bijection is > > if one clings to the idea that all natural numbers are finite, which is > > impossible. > > Back on your horse again. Tell me about the binary numbers (extended to the > left with 0's) where the leftmost 1 is in a finite position. Are all those > numbers finite? Are there only finitely many of them? > yes and yes -- Smiles, Tony |