An Ultrafinite Set Theory
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From: Aatu Koskensilta on 3 Mar 2010 11:37 MoeBlee <jazzmobe(a)hotmail.com> writes: > As far as I know, first order PA is not adequate. (I'm open to being > convinced otherwise, though.) Sink your teeth into the mumblings of Feferman on predicativism, ACA_0, proof-theoretic reductions, what not, then! -- 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 11:41 On Mar 3, 10:37 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > As far as I know, first order PA is not adequate. (I'm open to being > > convinced otherwise, though.) > > Sink your teeth into the mumblings of Feferman on predicativism, ACA_0, > proof-theoretic reductions, what not, then! But isn't ACA_0 a theory not compatible in strength with PA rather than a theory weaker or equal to PA? MoeBlee
From: MoeBlee on 3 Mar 2010 11:41 On Mar 3, 10:41 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > compatible I meant 'comparable'. MoeBlee
From: Aatu Koskensilta on 3 Mar 2010 11:42 MoeBlee <jazzmobe(a)hotmail.com> writes: > But isn't ACA_0 a theory not compatible in strength with PA rather > than a theory weaker or equal to PA? Nope. ACA_0 aka elementary analysis is a conservative extension of PA. -- 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 11:59
On Mar 3, 10:42 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > But isn't ACA_0 a theory not compatible in strength with PA rather > > than a theory weaker or equal to PA? > > Nope. ACA_0 aka elementary analysis is a conservative extension of PA. Okay, conservative extension. I don't know how that would leave my conjecture that PA is not enough for analysis. I guess it would depend on how that conjecture is more exactly stated. But still, it seems something must be added to PA (even if only by conservative extension) to get analysis cooking? No? MoeBlee |