FOM: Truth of G

Arnon Avron aa at
Thu Nov 9 02:50:33 EST 2000

Panu Raatikainen wrote:
> 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.
What the pure proof-theoretical argument demonstrates is not G,
but Con(T)->G. If one wants to go on and prove G (more accurately,
that G is *True*, because G cannot be proved in T) then I dont see 
how this can be done without
a reference to the standard model of T. First, the claim that
G is true already refers to the standard model (it is not true in all
models of T). Second, con(T) is *not* the claim that 
"one cannot derive from the axioms of T, by the chosen rules of inference,   
B & not-B (for some sentence B)". Con(T) is a certain formal formula
in the language of T (i.e. a formula about a property of the 
operations of + and *, which they have in some models of T and lack
in others). Only under a certain interpretation it is true in the
*standard* model iff T is consistent. One may say perhaps that con(T)
is not only true (in the standard model) iff T is consistent, but
in fact expresses this consistency. Even so, it only expresses this
in the standard model, and if we try to ignore this model we have 
no ground to accept con(T) or to connect it in any way to the consistency
of T. In such a case we have no ground to accept G or to claim
that we have a proof it.  

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