An Ultrafinite Set Theory
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From: Frederick Williams on 8 Mar 2010 21:24 RussellE wrote: > > On Mar 8, 9:33 am, Frederick Williams <frederick.willia...(a)tesco.net> > wrote: > > Transfer Principle wrote: > > > > > [...] But of course, as I myself found > > > out, it's far easier to give a theory with both finite and > > > infinite models than it is to give one with arbitrarily large > > > finite models but no infinite models. > > > > Compactness forbids it. > > Interesting. Would you please expnad on this? Compactness says that a set of first order sentences has a model iff every finite subset of it has a model. It follows from the completeness theorem. -- I can't go on, I'll go on.
From: RussellE on 8 Mar 2010 22:26 On Mar 8, 5:27 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Mar 8, 5:40 pm, RussellE <reaste...(a)gmail.com> wrote: > > > On Mar 8, 9:33 am, Frederick Williams <frederick.willia...(a)tesco.net> > > wrote: > > > > Transfer Principle wrote: > > > > > [...] But of course, as I myself found > > > > out, it's far easier to give a theory with both finite and > > > > infinite models than it is to give one with arbitrarily large > > > > finite models but no infinite models. > > > > Compactness forbids it. > > > Interesting. Would you please expnad on this? > > Interesting? I thought you weren't interested in mathematical logic. I'm not. That is why I read sci.math. > It's a theorem that if a theory has arbitrarily large finite models > then it has an infinite model. The proof is available in virtually any > textbook on mathematical logic. Would this forbid a theory which doesn't allow arbitrarily large models? I know the OP was refering to TP's attempts to find a theory which allow such models. Russell - Mathematics is the only true religion
From: RussellE on 8 Mar 2010 22:42 On Mar 8, 3:50 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <92c8f83f-5156-493d-af02-407e091a7...(a)s36g2000prf.googlegroups.com>, > > What are you left with, if, like me, you think > > infinite sets are contradictory? The only > > idea I can come with is natural numbers > > can not grow without limit and there exists > > a "largest" natural number. > > Insisting on a "largest possible natural" in a supposedly expanding > universe seems a bit contrary. I have to agree with you. Maybe Zeno was right and change is an illusion. > > > > One thing I do like about Raatikainen's > > paper is the possibility of proving the > > consistency of a finite theory. > > > I find it amazing so many people spend > > so much time and effort on systems like > > ZFC which could be proven inconsistent > > at any time. I would think we want to start > > with a theory that is provably consistent. > > What good would provable consistency be if your provably consistent > system should impose restrictions making much of current mathematics > impossible? It could lead to better mathematics. > So until someone actually proves those set theories with infinities now > extant to be inconsistent, I prefer to keep them. Assuming the earth is flat is a good approximation unless you are mapping the orbits of heavenly bodies. Russell - Zeno was right. Motion is impossible.
From: Virgil on 8 Mar 2010 23:47 In article <68c681ac-e945-4bdb-8185-8d5eef4c81cb(a)b5g2000prd.googlegroups.com>, RussellE <reasterly(a)gmail.com> wrote: > > What good would provable consistency be if your provably consistent > > system should impose restrictions making much of current mathematics > > impossible? > > It could lead to better mathematics. And would all the bridges built using the old math fall down? > > > So until someone actually proves those set theories with infinities now > > extant to be inconsistent, I prefer to keep them. > > Assuming the earth is flat is a good approximation > unless you are mapping the orbits of heavenly bodies. Assuming the Earth is flat would make most skiing pointless.
From: FredJeffries on 9 Mar 2010 09:23
On Mar 8, 7:26 pm, RussellE <reaste...(a)gmail.com> wrote: > On Mar 8, 5:27 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > It's a theorem that if a theory has arbitrarily large finite models > > then it has an infinite model. The proof is available in virtually any > > textbook on mathematical logic. > > Would this forbid a theory which doesn't allow arbitrarily large > models? > > I know the OP was refering to TP's attempts to > find a theory which allow such models. > See Donald Knuth's "Mathematics and Computer Science: Coping with Finiteness", Science 17 December 1976: Vol. 194. no. 4271, pp. 1235 - 1242 Available at www.sciacchitano.it/Spazio/Coping%20with%20Finiteness.pdf "Finite numbers can be really enormous, and the known universe is very small. Therefore the distinction between finite and infinite is not as relevant as the distinction between realistic and unrealistic" |