FOM: Truth of G

[Stephen Fenner]

On Tue, 7 Nov 2000, Raatikainen Panu A K wrote:

> The structure of Goedel's proof is (very roughly) the following:
>
> ASSUME THAT (e.g.)  PA  is consistent (otherwise it proves
> EVERY sentence and is trivially complete).
> By diagonalization, one can construct a sentence G that is
> independent (neither provable nor refutable) of PA.
>
> So far so good, but how,  then, can one conclude that G is true ?
> There are even two somewhat different ways...
>
> First, assuming that the provability predicate used is normal, one
> can show (and prove even inside PA) that
>         G <->  Cons(PA).
> (Although one can prove neither side of the equivalence in PA).
> Therefore, the truth of the sentence is, in a sense, already
> assumed in the beginning of the proof.
>
> NOTE: No non-mechanical intuition, no use of the standard model,
> is involved.

Here is where I lose you.  I think the original hypothesis, stated in the
metalanguage "Assume PA is consistent...." is always taken to mean,
"Assume that there is no _standard_ proof of 0=1 in PA" and so the
standard model is implicitly assumed.  If PA is consistent, then
PA + ~Con(PA) is also consistent and thus has a (nonstandard) model.  In
this model, G is false.

Of course, if you fix a model of PA and interpret Goedel's theorem in this
model, then you are only saying something nonvacuous if the model
satisfies Con(PA) to begin with.  This makes your statement above
plausible, but I don't think most people think of Goedel's theorem this
way.

Stephen Fenner
Computer Science and Engineering Department
University of South Carolina