Sorry for being unclear, Andy. What I meant by "existence of a truth-value for P within PA" is "definability within PA of a number whose value is 1 if P and 0 if not-P". Now that I think about it, I suspect that for every P in PA, there's a way to define (within PA) a number n such that "n=1 iff P" and "n=0 iff not-P" are both provable in PA. But it's 1:00am, so I don't put much faith in what I think or suspect right now. Jim On Wed, Sep 4, 2013 at 12:37 AM, Andy Latto <andy.latto@pobox.com> wrote:
On Tue, Sep 3, 2013 at 5:46 PM, James Propp <jamespropp@gmail.com> wrote:
(Here I'm ducking the issue of finding a specific proposition P such that the existence of a truth-value for P is provable in PA even though P and not-P are undecidable in PA, since it's not germane to my point here, which is that the mathematical question that I raised doesn't really capture my original motivating intuition.)
I'm not sure what you mean by "existence of a truth value" for P is. But if you talk about provability, rather than truth, this can't be done. Given a proposition P such that
"Either P or not-P is provable" is provable, then either P is provable, or not-P is provable. And if P is provable, so is "P is provable", because there exists an N such that N is the goedel number of a proof of P, and a statement of the form "N is the goedel number of a proof of P" is always provable if true.
Andy
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