Obections to Cantor's Theory (Wikipedia article)
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From: Robert Low on 29 Jul 2005 04:50 Han de Bruijn wrote: > I know that. But I'm changing the rules of the game in accordance with > the mantra: > A little bit of Physics would be NO Idleness in Mathematics > Then {} is a member of {} . OK, Han. Let's try your game. What are the axioms you use in place of ZF(C)? Once you've listed them, tell us what your standard model is (an equivalent for the cumulative hierarchy), and what the natural numbers are in that standard model.
From: Han de Bruijn on 29 Jul 2005 05:03 Virgil wrote: > Prime example of that is classical number theory. G. H. Hardy was sure > that his work in that area could not ever be put to any practical use, > but it has turned out to be the foundation for coding schemes imperative > for the security of today's electronic commerce. You keep on saying this. But you forget one thing. The idealization of numbers has been a very _good_ one. This means that you can stay, safe and well in your ideal world, without bothering whether yes or no your work could ever be put to any practical use. But we are not questioning number theory, (Euclidian) geometry, calculus or linear algebra here. We are questioning Set Theory. And that is quite a different matter. Han de Bruijn
From: Han de Bruijn on 29 Jul 2005 05:07 Martin Shobe wrote: > Maybe I should have said "Set theory does not have physics as it's > immediate inspiration, so failing to satisfy some direct connection to > physics is irrelevant." No. It's highly relevant. The fact that something IS such and so doesn't mean that it SHOULD BE such and so. (Maybe some shouting helps ...) Han de Bruijn
From: Han de Bruijn on 29 Jul 2005 05:14 Martin Shobe wrote: > On Thu, 28 Jul 2005 15:56:32 +0200, Han de Bruijn > <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>But without the claim that it >>is the one and only foundation possible. Why not have _several_ pillars >>that provide a foundation, instead of just one? > > Actually, I don't have a problem with that. In a sense, we have that > now with set theory and category theory (And lets not forget logic). > But while physics will continue to provide inspirition to mathematics, > it will not qualify as a foundation for methematics. And I don't want that either. Read my lips: A little bit of Physics would be NO Idleness in Mathematics See? Just that tiny pinch of salt in your otherwise tasteless soup. But nevertheless: *IN* your soup. Han de Bruijn
From: Han de Bruijn on 29 Jul 2005 05:19
Robert Low wrote: > Martin Shobe wrote: > >> Maybe I should have said "Set theory does not have physics as it's >> immediate inspiration, so failing to satisfy some direct connection to >> physics is irrelevant." > > I was just being picky...and besides, the 'connection to physics' > that set theory 'fails to satisfy' is almost entirely obscure > to me. I see no physical argument to justify a={a}. Does {} belong to the set? No? Then the element "an sich" is the same as the set which contains only that element. Physically speaking. Han de Bruijn |