An Ultrafinite Set Theory
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From: Frederick Williams on 9 Mar 2010 10:42 RussellE wrote: > > 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? A first order theory that has arbitrarily large models has an infinite model. Do you want small finite models or no finite models at all? -- I can't go on, I'll go on.
From: Frederick Williams on 9 Mar 2010 10:52 RussellE wrote: > > On Mar 8, 3:50 pm, Virgil <Vir...(a)home.esc> 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. Eh? Impossible means better? In any case if a "better" mathematics is required isn't it for mathematicians to ask for it and supply it? > > 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. And numerous other things such as sailing across the atlantic. -- I can't go on, I'll go on.
From: MoeBlee on 9 Mar 2010 11:16 On Mar 8, 9:26 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? First, to be clear, I should have mentioned that I'm referring to first order theories in this context. (So, in this context, when I say 'theory', that is short for 'first order theory'). Now, to answer your qutestion: If a theory has an infinite model, then it has arbitrarily large infinite models. And, if a theory has arbitrarily large finite models, then it has an infinite model, so it has arbitrarily large infinite models. On the other hand, there are theories that only have models less than a certain finite cardinality, so those theories don't have arbitrarily large models. MoeBlee
From: MoeBlee on 9 Mar 2010 11:19 On Mar 8, 9:26 pm, RussellE <reaste...(a)gmail.com> wrote: > 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. So what? Mathematical logic is mathematics. Anyway, at least to the extent of this particular question about the cardinality of models, you've just shown that you are interested in mathematical logic. MoeBlee
From: bobg0 on 9 Mar 2010 15:55
On Mar 7, 8:16 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > And so the task remains. What can I do to the "crank" theories in > order to make them "more powerful" and "less work to use," so that > they'd be more palatable to the standard theorists? > > For starters, "less work to use" implies that we should minimize the > number of axioms, schemata, primitives, etc., required to use it. In > another thread, a theory I wrote about the "crank" ellipsis had over > a score of axioms. By contrast, ZFC is usually written as having only > about 7-10 axioms (counting schemata as single axioms). And so I > must cut "crank" theories down so that they fall near this range. According to your standards then the mathematics of Eudoxus is less palatable than that of Pythagoras, the geometry of Riemann is less palatable than that of Euclid, the mechanics of Newton less palatable than that of Aristotle and Einstein's less palatable than Newton's, constitutional democracy less palatable than absolute monarchy, a computer less palatable than a tally stick, an arc welder less palatable than a stone hatchet,... |