An Ultrafinite Set Theory
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From: MoeBlee on 8 Mar 2010 20:27 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. 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. MoeBlee
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. |