Obections to Cantor's Theory (Wikipedia article)
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From: Barb Knox on 19 Jul 2005 21:54 In article <7mrqd1liuvhopdetv1mfe7ii498hru8qr4(a)4ax.com>, G. Frege <nomail(a)invalid> wrote: >On 19 Jul 2005 21:02:29 GMT, Chris Menzel <cmenzel(a)remove-this.tamu.edu> >wrote: > >> >> Pretty clearly, you aren't terribly well-educated in set theory. Don't >> you think you should understand a field before you try to point out its >> flaws? >> >Isn't that rather typical for a m o d e r n "anti-cantorian"? :-) I am the very model of a modern non-Cantorian, With insights mathematical as good as any saurian. I rattle the Establishment foundations with prodigious ease, And populate the counting numbers with some new infinities. I've never studied axioms of sets all theoretical, But that's just ted'ous detail; whereas MY thoughts are heretical And cause the so-called experts rather quickly to exasperate, While I sit back and mentally continue just to .... -- --------------------------- | 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: Virgil on 19 Jul 2005 22:03 In article <MPG.1d471b60f909ffa2989f28(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > David Kastrup said: > > 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. > I still do not get this. You have a set of naturals {0,1,2,3...}, and > a set of binary numbers {0,1,10,11,100,101,110,111,....}. Surely > there is a bijection between these two sets. So, what is the problem > if one interprets the binary numbers (with implied leading zeroes) as > being a map of each subset, where each successive bit represents > membership in thesubset by each successive natural number? TO's mapping only covers finite subsets of N, but leaves uncovered every infinite subset, including N itself. For example there is no natural in TO's mapping that maps to the set of odds, nor to the set of evens, nor to the set of primes, and so on. > > I mean, your statement that begins with "Clearly" is not at all clear > to me. Why such things remain unclear to TO is something more appropriate to analysis by psychologists than mathematicians.
From: Virgil on 19 Jul 2005 22:08 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.
From: Dik T. Winter on 19 Jul 2005 22:09 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. -- 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 22:16
In article <MPG.1d472150e32f79c5989f2a(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > David Kastrup said: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > In all actuality, the flaws in various proofs and assumptions in set > > > theory have been directly addressed, and ignored by the mainstream > > > thinkers here. > > > > Guffaw. > Again with the insulting remarks. Grow up. Such reactions to such stupidity/ignorance are almost inevitable. > > Assume a set X can be put into complete bijection with its powerset > > P(X) such that we have a mapping x->f(x) where x is an element from X > > and f(x) is an element from P(X). Now consider > > Q = {x in X|x not in f(x)}. Clearly, for all x in X we have > > Q unequal to f(x), since x is a member of exactly one of f(x) and Q. > > So Q is missing from the bijection. > > > > > Again with the "Clearly". You might want to refrain from using the word, and > just try to be clear, without hand-waving. No handwaving required, just clear thinking. If TO cannot think that clearly, it is hardly the fault of the logic. |