FOM: Hartry Field's argument concerning "determinate truth"
Stephen G Simpson
simpson at math.psu.edu
Tue Mar 3 18:47:51 EST 1998
Charles Silver 27 Feb 1998 09:27:54 said:
> >If Goldbach's Conjecture were proven to be formally undecidable in PA, it
> >must then be true.
Hartry Field 27 Feb 1998 12:30:39 replied:
> If 'finite' and 'natural number' are determinate, then the argument
> is correct. If they are indeterminate, then the argument is not
> correct, for the truth of all the instances involving "genuine
> natural numbers" doesn't guarantee the determinate truth of the
> universal quantification over all numbers.
Field also followed up on this in a posting of 2 Mar 1998 08:57:14,
explaining his notion of "determinate truth" in terms of "provability
in our strongest accepted theory M".
I still don't understand Field's point. Maybe I'm just being dense.
It seems to me that Silver's argument shows the following: If T is any
of a wide range of theories, and if T proves that GC is undecidable in
PA, then T proves GC. This holds for T = ZFC, or T = true arithmetic,
or T = M. Each of these special cases is interesting. In the case of
true arithmetic, we get: If GC is *in fact* undecidable in PA, then GC
is *in fact* true. In the case of T = M, we get: if it is
"determinately true" that GC is undecidable in PA, then GC itself is
"determinately true". In other words, Silver's argument is correct
even if we interpret everything in terms of M. So I don't see how
Field's notion of "determinate truth" has any bearing on this. What
am I missing? Maybe Field is really agreeing with me and I'm just not
understanding some other subtle point that he is making.
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