[FOM] Proof "from the book"
Arnon Avron
aa at tau.ac.il
Tue Aug 31 04:33:39 EDT 2004
>
>
> Arnon Avron says:
>
> >A Godel sentence (NOT "the"
> >Godel sentence) for a consistent extension of Q is true, and
> >Godel's proof does show this
>
> Godel's proof does indeed show the implication "if S is consistent then
> G is true", which is provable in S itself. I don't think you mean to
> say that if S is consistent, Godel's proof shows G to be true.
That "if S is consistent then G is true" is provable not only in S (which
might prove false sentences), but also in PA (which proves only true
sentences). Hence I do say that if S is a (formal) consistent extension of Q
then Godel's proof shows G to be true.
One more note: I assume here of course that an absolute notion of
truth exists for some propositions concerning finite objects like natural
numbers or proofs in a formal system. Without assuming this, it makes
no sense to talk about what Godel's proof proves (intuitionists will
object perhaps, but I admit that I was never able to understand how
can they coherently believe in what they claim they believe). Needless
to say, all theorems of PA (to say nothing about Q) are TRUE.
Arnon Avron
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