FOM: Simpson's query on my response to Silver

hartry field hf18 at
Mon Mar 2 08:57:14 EST 1998

I intended to be commenting not on an argument for the claim (1) that if IT
IS PROVABLE THAT Goldbach's Conjecture is undecidable in PA then it must be
true, but on an argument for (2) that if GC  IS undecidable in PA then it
must be true.  (Looking back at Silver's posting, I see that he did state
the conclusion of the argument as (1).  However, the argument offered was in
fact an argument for (2), and I think this must have been what he meant.)

The hypothesis I was entertaining (and maintaining the coherence of, though
not endorsing) was that there is no more determinacy to the concepts of
'finite' and 'natural number' than the most powerful recursively axiomatized
first order mathematical theory we accept gives them; so that sentences
undecidable in this theory M (or more exactly, in all reasonable candidates
for it--this qualification is necessary because of the vagueness of
'accept') lack determinate truth value.  If GC is undecidable in PA, it may
well be undecidable in (all reasonable candidates for) M, so it might on the
hypothesis entertained lack determinate truth value.  I took the subtext of
Silver's argument to be an objection to the coherence of this.  The response
of course is that though the undecidability of GC shows its truth in the
standard model, it may not be true in all models of (all candidates for) M.

(As I noted,) an added wrinkle is that 'undecidable' also admits nonstandard
interpretations, so it might be indeterminate whether GC is undecidable (on
the hypothesis being entertained).  But my point is that undecidability in
the standard model doesn't entail determinate truth.  Or if you like, lack
of determinate decidability doesn't entail determinate truth.  So the Silver
argument can't be used against the hypothesis I was entertaining.  (Perhaps
Silver didn't intend the argument to be so used, but that's what his
posting, in the context of Franzen's, seemed to me to suggest.)

Turning to (1), I agree that if we could prove, in some theory M that we
accepted, the PA-undecidability of GC (and couldn't also prove the
contrary!), then that would be good reason to regard GC as determinately
true.  But that would be because this constituted in effect an M-proof of
GC.  So it isn't contrary to the idea that determinate truth coincides with
provability in "our maximal theory". 

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