Obections to Cantor's Theory (Wikipedia article)
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From: Han de Bruijn on 21 Jul 2005 07:34 David Kastrup wrote: > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > >>Daryl McCullough wrote: >> >>>Nothing of any importance about mathematics would change if we >>>substituted different words for the basic concepts. In contrast, >>>your arguments are about nothing *but* terminology. To me, that >>>shows that there is no actual content to your arguments. An actual >>>mathematical argument does not depend on word choice. As a >>>challenge, see if you can express your claims about infinite sets, >>>or infinite naturals, or set size, or whatever, *without* using the >>>words "infinite", "larger", "size", etc. >> >>This clearly represents the formalist (Hilbertian) view on >>mathematics as a "senseless game with symbols". > > So, it represents the view that it does not make sense to talk about > different things using the same words. No. You're twisting my words and those of Daryl McCullough. (Though I feel much more comfortable with your interpretation than with his.) No trouble with the rest of your writeup [ deleted ]. Han de Bruijn
From: Han de Bruijn on 21 Jul 2005 07:37 David Kastrup wrote: > The rules are: if you don't like some conclusions, you have to change > the axioms, and then you lose all other conclusions (many of them > might be easy to reacquire, but that process is not automatic). But what if your method is not axiomatic ? I mean, in intuitionism, the emphasis is not on formal reasoning and axions, but "constructiveness". Han de Bruijn
From: Peter Webb on 21 Jul 2005 07:41 >> >>> I will have my web pages published before too long, so I am not >> >>> getting into a mosh pit with you again right now. Just be aware that >> >>> anti-Cantorians are sick of being called crackpots, and the day will >> >>> soon come when the crankiest Cantorians will eat their words, and >> >>> this rot will be extricated from mathematics. >> >> > I never claimed you would be cast from the ranks of mathematics, but that > you > will see the errors that you are currently ignoring, and that the rot of > Cantorian cardinality will be removed from mainstream thought and replaced > with > ideas that don't lead to absurdity like Banach-Tarski. I do see the > ramifications of this nonsense in many areas. Until you can demonstrate > that > the theory is really correct, Correct? What does that mean? Really, I have no idea what the word means in this context. Inconsistent perhaps? If so, how? >I am well within my rights to disagree with your > axioms and conclusions, How do you disagree with an axiom? How about Group Theory, for example? Do you disagree that there is element e such that g*e=g for every g? What does it mean that you disagree with an axiom? Is it that you don't think that there is a model which satisfies the axiom? If so, are you contending that N doesn't model PA? Could you explain exactly which axiom(s) you "disagree with", what "disagree with" means, and why you "disagree with" them? We can then move on to conclusions, where I don't understand what "disagree with" them means either. > and if that right is challenged, I will continue to > defend it and challenge your theory. It takes two to tango, and if you end > up > with people vowing vengeance, well hell, you probably deserve it. Then > again, > maybe JSH is mentally unstable, but then so was Cantor, and so was Godel. > -- You might think that infinite ordinals and cardinals are "absurd" and "nonsense". You are in good company historically. Lots of people thought that zero was absurd and nonsense, as were negative numbers (show me -3 cows), imaginary numbers, irrational numbers, positional notation ... of course, 100 years after each of these absurd and nonsensical concepts were introduced, it was only the cranks that continued to rail against them. Its like somebody saying 100 years after Copernicus that the geocentric model leads to absurdities and obvious nonsense like the earth is spherical, when it is obviously flat. This would be likely to lead to the same type of contempt from astronomers as you are getting from the mathematicians here.
From: Martin Shobe on 21 Jul 2005 07:53 On Thu, 21 Jul 2005 12:21:38 +0200, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >stephen(a)nomail.com wrote: > >> These seems to be another common misconception among >> the anti-Cantorians that words cannot have specific >> meanings in specific contexts. Somehow they >> think an all-encompassing definition of 'infinite' must >> be provided before someone can say what an infinite set is. >> I am not sure what they mental hangup is. I wonder >> how any of them would ever learn a foreign language. > >Ha, ha, ha. _This_ anti-Cantorian has learned six languages: Dutch, >German, French, English, Latin and Greek. We in the Netherlands are >privileged with our knowledge of foreign languages. Yet I find that >an "all-encompassing definition of 'infinite' must be provided". Does a "Christmas tree" in certain engineering contexts actually have be a conifer? Martin
From: Dik T. Winter on 21 Jul 2005 07:55
In article <MPG.1d48369d8fca2174989f42(a)newsstand.cit.cornell.edu> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > 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 I think you should apply for the reward for solving Collatz' problem. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |