Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 20 Jul 2005 21:52 In article <MPG.1d4858812235c0b0989f4c(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > David Kastrup said: > > What is supposed to be a "constant equality"? > An equality that holds true for n=1, and for n=n+1 given true for n. In mathematics, n=n+1 is always false.
From: Virgil on 20 Jul 2005 21:56 In article <MPG.1d485a8d648b2c6f989f4d(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > >>> I offered, and you saw, a deductive proof that proves that the > > >>> largest natural in a set must be at least as large as the set > > >>> size. > > > > Yes. So we have: > > > > If S is a set of natural numbers, and S has a largest member N, > > then N >= the cardinality of S. > > > > How do you prove that every set of natural numbers has a largest > > member? > > > > -- Daryl McCullough Ithaca, NY > > > > > Essentially, the proof shows that no set can have a larger number of > naturals in it than the values of all the naturals in it. This is nonsense. "A larger number of naturals in it than the values of all the naturals in it" makes no sense. TO performing as usual.
From: Dik T. Winter on 20 Jul 2005 22:12 In article <MPG.1d483583ff4dfb97989f41(a)newsstand.cit.cornell.edu> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: .... > > 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. .... > 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. Actually that is the only conclusion. But that conclusion can be shown to be equivalent to other conclusions. > 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, Yes, you have told that already numerous times without actually showing that is true. But let us go from the binary to the dyadic notation. In that case the digits used are 1 and 2. 0 can not be represented, but each natural number can be represented in only *one* way by a string of those digits. A short table to show the idea: 1 = 1 2 = 2 3 = 11 4 = 12 5 = 21 6 = 22 7 = 111 8 = 112 9 = 121 10 = 122 etc. In this representation each finite natural number is represented by a single finite string. Now how many of such finite strings are there, given that the stringlength is unbounded? How would your "number" "N" be represented in that representation? > and the > conflation of "larger and infinite" with "uncountable" is arbitrary. You are missing the point. Each set "larger" in some sense than the naturals as mathematicians think about them is (by definition) uncountable. > The > diagonal proof has hidden assumptions that mathematicians seem to > acknowledge and not question, Wrong, the diagonal proof proves the same thing as the proof that the powerset of a set is larger than the base set. The reals can be put in a 1-1 relation with the powerset of the natural numbers. (But only when you consider natural numbers in the sense of the mathematicians.) -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 20 Jul 2005 22:17 In article <MPG.1d4840c473d9f90d989f4a(a)newsstand.cit.cornell.edu> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > Han de Bruijn said: > > David Kastrup wrote: > > > > > Oh good grief. Successor in interest to JSH, are we? > > > > JSH is not an anti-Cantorian. So this argument, again, doesn't make > > sense and may only be useful for the purpose of insulting. > > Yes, they love to hold up JSH as the ultimate crank, and apparently > consider me as some kind of contender. In a way you are when you state that in 10 years time your view will be winning and standard mathematics will crumble in dust. That is just about the same as JSH has written. And that was the remark David Kastrup made his remark on. There are more ways in which you are similar. You both refuse to study the base material from textbooks. You both refuse to use standard terminology... -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 20 Jul 2005 22:37
In article <MPG.1d487b3445920e87989f55(a)newsstand.cit.cornell.edu> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > Chris Menzel said: .... > > If the assumption you are referring to is that there is a list of reals > > of length omega_1 (= aleph_1 in pure set theory), then all you are doing > > by denying that this assumption requires the continuum hypothesis is > > showing your abject ignorance of the subject. > > The continuum hypothesis states, if I'm not mistaken, that there are no > infinities between aleph_0 and aleph_1. You are mistaken. By definition "aleph_1" is the smallest cardinal larger than "aleph_0", so there are no cardinals between "aleph_0" and "aleph_1" by definition. The continuum hypothesis states that the cardinality of the reals is "aleph_1". Ignorance I would argue. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |