Torkel Franzen on truth
in [Logic]
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From: LauLuna on 10 Nov 2007 05:00 On Nov 9, 10:47 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >He clearly believes that the we know PA/ZFC to be consistent with the > >same certainty as any mathematical theorem i.e. we can prove G. So the > >question arises how we can construct a machine that can do the same. > >obviously not by emulating PA/ZFC. > > Well, here's an attempt at describing an informal metatheory that > captures a lot of human metatheoretic reasoning: > > 1. Every axiom of ZFC is true. > > 2. For every statement Phi in the language of ZFC, > Phi <-> Phi is true. > > 3. If T is any theory in the language of ZFC, and every > axiom of T is true, then every theorem of T is true. > > This informal theory can prove Con(ZFC) and > Con(ZFC + Con(ZFC)), etc. And it's all perfectly > mechanical; you can write a program to work out > all the consequences of rules 1-3. > > Of course, we can give a name to this new theory: > > Let ZFC_1 = the collection of all statements in > the language of ZFC that follow from rules 1-3. So defined, ZFC_1 is not the informal theory you described, since there is no predicate in the language of ZFC expressing the truth predicate for ZFC sentences, by Tarski's indefinability theorem. Regards
From: Newberry on 10 Nov 2007 11:53 On Nov 9, 4:17 am, aatu.koskensi...(a)xortec.fi wrote: > On 9 Oct, 10:20, Peter_Smith wrote: > > > Read "pretty good reason" to mean "at least pretty good reason, maybe > > conclusive reason". As it happens I think there are conclusive reasons > > to believe PA consistent. > > Yes, PA is obviously consistent. Con(T) --> T(G) --> G We have just proven G. Now there are two possibilities a) The human mind surpasses any machine b) The human mind does not surpass a machine TF is in favor of b. In the first case the human mind can see the axioms of PA as manifestly true, which a machine cannot. If b is the case then I wonder how we can construct a machine that can generate all the truth of PA.
From: Newberry on 10 Nov 2007 11:58 On Nov 9, 1:47 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >He clearly believes that the we know PA/ZFC to be consistent with the > >same certainty as any mathematical theorem i.e. we can prove G. So the > >question arises how we can construct a machine that can do the same. > >obviously not by emulating PA/ZFC. > > Well, here's an attempt at describing an informal metatheory that > captures a lot of human metatheoretic reasoning: > > 1. Every axiom of ZFC is true. > > 2. For every statement Phi in the language of ZFC, > Phi <-> Phi is true. > > 3. If T is any theory in the language of ZFC, and every > axiom of T is true, then every theorem of T is true. > > This informal theory can prove Con(ZFC) and > Con(ZFC + Con(ZFC)), etc. And it's all perfectly > mechanical; you can write a program to work out > all the consequences of rules 1-3. > > Of course, we can give a name to this new theory: > > Let ZFC_1 = the collection of all statements in > the language of ZFC that follow from rules 1-3. > > Then we can come up with yet another theory by > modifying rule1: > > 1'. Every axiom of ZFC_1 is true. > > Then we could let ZFC_2 be the set of all consequences > of rules 1', 2, and 3. etc. Are ZFC_1, ZFC_2 etc. consistent? How do we know that they are? Can a machine generate theories ZFC_1, ZFC_2 etc?
From: Peter_Smith on 10 Nov 2007 15:52 On 10 Nov, 16:53, Newberry <newberr...(a)gmail.com> wrote: > b) The human mind does not surpass a machine > > TF is in favor of b. > > If b is the case then I > wonder how we can construct a machine that can generate all the truth > of PA. Why should any machine be able to generate all the truths of (the language of) PA? After all, we can't do that either.
From: Newberry on 10 Nov 2007 17:38
On Nov 10, 12:52 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > On 10 Nov, 16:53, Newberry <newberr...(a)gmail.com> wrote: > > > b) The human mind does not surpass a machine > > > TF is in favor of b. > > > If b is the case then I > > wonder how we can construct a machine that can generate all the truth > > of PA. > > Why should any machine be able to generate all the truths of (the > language of) PA? After all, we can't do that either. How can we construct a machine that can generate all the truth of PA that we can? |