Torkel Franzen on truth
in [Logic]
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From: Daryl McCullough on 10 Nov 2007 19:03 Newberry says... > >On Nov 9, 1:47 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> 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? They are consistent if ZFC is true. >How do we know that they are? I can't say for sure I do know that they are, but some people might. >Can a machine generate theories ZFC_1, ZFC_2 etc? Sure. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 10 Nov 2007 19:05 LauLuna says... > >On Nov 9, 10:47 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> 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. That's why I said "all statements in the language of ZFC" rather than "all statements". -- Daryl McCullough Ithaca, NY
From: Peter_Smith on 10 Nov 2007 19:20 On 10 Nov, 22:38, Newberry <newberr...(a)gmail.com> wrote: > How can we construct a machine that can generate all the truth of PA > that we can? Well, who knows which truths *those* are?
From: Newberry on 10 Nov 2007 22:05 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. > OK, how do we reconcile it with this? >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55
From: Peter_Smith on 11 Nov 2007 03:29
On 11 Nov, 03:05, Newberry <newberr...(a)gmail.com> wrote: > 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. > > OK, how do we reconcile it with this? > > >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55 There is no conflict at all between what I said (something TF held too), and that latter quote. To hold that PA is clearly consistent is quite compatible with holding that, with some arbitrarily thrown- together extension of Q, we won't in the general case know whether it is consistent, and hence won't know whether its canonical Gödel sentence is true. |