[FOM] Godel Sentence

[Richard Heck]

On Mon, 2003-08-25 at 04:40, Arnon Avron wrote:
> First and most impoerant: In my previous posting I asked for references 
> to works on the following question:
> 1) How should the general concept of "A Godel sentence for a
> theory T" be defined in exact, mathematical terms (so the question whether a given 
> sentence is a Godel sentence will have a clear-cut answer,
> even if we might not know that answer).

Here's an idea. Probably not a very good one, but an idea nonetheless,
and perhaps a starting point.

Say that G is a Goedel sentence for a theory T just in case there is
some "nice" (birecursive?) mapping *x* between sentences of the language
of T and terms of the language of T such that:
(i) G is a sentence of the language of T;
(ii) G has the form ~(Ex)(Bew(x,t)), where Bew(x,y) is a formula and t
is a term;
(iii) T proves: t=*G*;
(iii) Bew(x,y) represents "x is a T-proof of y" under the coding *x*.
I'm assuming here that proofs can be coded as sequences of sentences. So
we'll need to assume that we've got a coding of sequences available, and
we'll presumably want to assume that it too is nice. If I remember
right, there's some notion of an acceptable something-or-other that is
probably what we need here. I can't remember that terminology or exactly
what the notion is. And my books are at the office. Someone?

Those certainly seem like simple necessary conditions. One might
consider adding something like:
(iv) (Ex)Bew(x,y) satisfies the derivability conditions.
For one might, for some purpose, want to know that Bew(x,y) not only
gets the extension of "x is a proof of y" correct but in some sense gets
its intension right, as well. Of course, it is controversial whether the
derivability conditions should be required for that purpose, so it will
be similarly controversial whether they should be included here.

> 3) Under what conditions are two different Godel sentences (obtained,
> e.g., by two totally different methods of coding), equivalent,
> and in what theory are they equivalent?

The answer to this question with the above conditions should be known to
people around here. I don't expect that everything that is a Goedel
sentence by just (i)-(iii) is equivalent in T, but I don't have a
counterexample. Then again, isn't the Rosser sentence a counterexample?
If not, something along those lines should be. On the other hand, if you
add (iv), then I believe all such sentences are provably equivalent, in
T, to Con(T), and so to one another. We may need to assume that T is
"strong enough", in the usual sort of sense.

Richard Heck