An Ultrafinite Set Theory
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From: Aatu Koskensilta on 3 Mar 2010 12:05 MoeBlee <jazzmobe(a)hotmail.com> writes: > Okay, conservative extension. I don't know how that would leave my > conjecture that PA is not enough for analysis. Your original comment was about adequacy for physics. Feferman has proposed that all mathematics needed, in a strictly logical sense, in physics can be formalized in theories proof-theoretically reducible to primitive recursive arithmetic. As for analysis, it may or may not be that PA is not enough, depending on just what we have in mind. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: MoeBlee on 3 Mar 2010 12:40 On Mar 3, 11:05 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > Okay, conservative extension. I don't know how that would leave my > > conjecture that PA is not enough for analysis. > > Your original comment was about adequacy for physics. Feferman has > proposed that all mathematics needed, in a strictly logical sense, in > physics can be formalized in theories proof-theoretically reducible to > primitive recursive arithmetic. As for analysis, it may or may not be > that PA is not enough, depending on just what we have in mind. Hmm, "proof-theoretically reducible". Here my ignorance in proof theory is hurting me. So, help me out, am I not right that "proof-theoretically reducible to PRA" is some distance from PRA itself? MoeBlee
From: Aatu Koskensilta on 3 Mar 2010 12:52 MoeBlee <jazzmobe(a)hotmail.com> writes: > So, help me out, am I not right that "proof-theoretically reducible to > PRA" is some distance from PRA itself? Consider two theories T and S. We say that T is proof-theoretically reducible to S w.r.t a class G of statements (e.g. arithmetical statements) if there's a recursive function f for which the following hold: 1. If x is (the code of) a proof, in T, of a statement A in the class G, then f(x) is (the code of) a proof of A in S. 2. The above (1) is provable in S. Feferman's argues that there are theories T in which we can develop all the mathematics needed (in a strictly logical sense) for physics that are proof-theoretically reducible to primitive recursive arithmetic (w.r.t arithmetical statements, and hence certainly any experimentally relevant statements). -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: MoeBlee on 3 Mar 2010 13:00 On Mar 3, 11:52 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Consider two theories T and S. We say that T is proof-theoretically > reducible to S w.r.t a class G of statements (e.g. arithmetical > statements) if there's a recursive function f for which the following > hold: > > 1. If x is (the code of) a proof, in T, of a statement A in the class > G, then f(x) is (the code of) a proof of A in S. > > 2. The above (1) is provable in S. > > Feferman's argues that there are theories T in which we can develop all > the mathematics needed (in a strictly logical sense) for physics that > are proof-theoretically reducible to primitive recursive arithmetic > (w.r.t arithmetical statements, and hence certainly any experimentally > relevant statements). Thanks. You've said 'in a strictly logical sense' a few times. Would you amplify what you mean by that in this context? MoeBlee
From: Michael Stemper on 3 Mar 2010 13:31
In article <bd7f0e77-5937-4896-b9fb-d676b9dcf9a6(a)o3g2000yqb.googlegroups.com>, MoeBlee <jazzmobe(a)hotmail.com> writes: >On Mar 2, 5:54=A0pm, RussellE <reaste...(a)gmail.com> wrote: >> On Mar 2, 3:33=A0pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> > RussellE <reaste...(a)gmail.com> writes: >> > > I have often been told there are no "consistent" ultrafinite set >> > > theories (UST). >> >> > Really? Have you been told so here on the newsgroup? >> >> > Can you point me to a single post in which someone said that? >> >> People have said that in this newsgroup (it might have been me). > >You "told" yourself then? I think that he's a Standard Theorist, attempting to suppress new ideas. They're very thorough in their suppression. -- Michael F. Stemper #include <Standard_Disclaimer> 91.2% of all statistics are made up by the person quoting them. |