Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 19 Jul 2005 23:05 In article <MPG.1d4736aa784eeb4f989f34(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > The World Wide Wade said: > > In article > > <MPG.1d472484f809e37c989f2c(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > The idea of uncountability as being equivalent to "larger than the set of > > > naturals" is unfounded. There is no reason to believe that larger sets > > > cannot > > > be enumerated. the power set of the naturals can be enumerated and > > > bijected > > > with the naturals, as I described in another post, as long as infinite > > > natural > > > numbers are allowed. > > > > Sort of like saying "if 0 = 1" is allowed. > > > No, more like saying "if infinite digits are allowed", which is what is > required to have an infinite set of digital numbers using a finite base. That > was a useless comment. I shouldn't even be responding. Neither infinite digits nor infinitely many digits are allowed in any STANDARD set of natural numbers. The things TO is playing with are of no use to anyone else, as they violate all sorts of rules that natural numbers must obey.
From: Virgil on 19 Jul 2005 23:08 In article <MPG.1d4736d25730cfe7989f35(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Stephen J. Herschkorn said: > > Should a reputable encyclopedia contain an entry devoted entirely to > > people who think the earth is flat? > > An entry only for those who think that sun revolves the earth? > > An entry devoted specifically to those who think that man never landed > > on the moon? > > To those who insist there is a smallest positive real number? > > > > > 000...000.000...001 It is not a real number unless there is a finite number of zeroes specified for each ellipsis. and once that is done, it is not a smallest positive real, since that is a mythical beast.
From: stephen on 19 Jul 2005 23:23 In sci.math Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: <snip> > Anyhow, Cantor's proof consists solely of logical conclusions from > antecedents. As two of the anti-Cantorians have stated to me, yes, > each step is right but the proof is wrong... > -- > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ It seems like a lot of the "anti-Cantorians" and other mathematical nay-sayers tend to start with their conclusions, and then try to work backward. The idea of starting with a fixed set of well defined axioms and working forward seems totally alien to them. Stephen
From: imaginatorium on 20 Jul 2005 00:48 Stephen J. Herschkorn wrote: > Should a reputable encyclopedia contain an entry devoted entirely to > people who think the earth is flat? I don't see why not. Wikipedia has no size constraints; if someone writes a lucid, comprehensive article on the history of flat-earthing, I think that might be extremely useful. No sensible person can keep up with the latest changes in the lunatic fringe - I still remember the real shock I experienced going downstairs in the Comp. Linguistics dept in Manchester and discovering that the latest conference being set up was of Believers in crystal healing. Pre-Internet, I think I would otherwise simply have dismissed such a thing as inconceivable. > An entry only for those who think that sun revolves the earth? > An entry devoted specifically to those who think that man never landed > on the moon? > To those who insist there is a smallest positive real number? Hmm. The first three of your putative beliefs are empirical facts about the world; the last isn't, and while the people claiming it may be unsophisticated cranks, in general, it is plausibly possible to study the mathematics of a system in which there is a smallest positive real number - the IEEExxxx floating-point system, for example. (This is fundamentally the problem with Orlovian maths, of course: everything means something different from usual, so one loses track or runs out of energy eventually.) Brian Chandler http://imaginatorium.org
From: G. Frege on 20 Jul 2005 00:54
On Wed, 20 Jul 2005 13:54:27 +1200, Barb Knox <see(a)sig.below> wrote: > > 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 .... > "I was asked that before, and never got around to fully analyzing the axioms for lack of time, but the diagonal proof suffers from the fatal flaw of assuming that the diaginal traversal actually covers all the numbers in the list." (Tony Orlow) F. |