FOM: Truth of G
Raatikainen Panu A K
Praatikainen at elo.helsinki.fi
Wed Nov 8 04:50:37 EST 2000
On 7 Nov 00, at 11:53, Stephen Fenner wrote:
> 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.
RE: My whole point was to emphasize that models need not have,
and did not in fact have, any role in Gödel's proof. It is a purely
proof-theoretical ("syntactical") construction. "Consistency of T", in
this context, means just that one cannot derive from the axioms of
T, by the chosen rules of inference, B & not-B (for some sentence
B). See, no models. Of course, people who accept the model talk
(in fact, I do) may then note that by Gödel's completeness
theorem, T has a model. But note that for many theories T, T may
not, as far as we know, even have a standard model. But the point
is that Gödel's proof makes perfect sense even for those who prefer
to avoid the very model talk and like to stick to proof theory.
Panu Raatikainen
Department of Philosophy
University of Helsinki
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