FOM: Truth of G

[Raatikainen Panu A K]

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