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george wrote:
> Aatu Koskensilta wrote:
> > For any extension
>
> formal extension
> > T of Robinson arithmetic we can effectively find a
> > sentence G_T, s.t. G_T is true
>
> true in the standard model
> > and undecidable in T
>
> The cart is somewhat before the horse here.
> If the sentence (G_T) is not in T, then by definition,
> there are BOTH models of T in which it is true AND
> models of T in which it is false. Since it is *T* that
> we are (formally/recursively) axiomatizing,
> discussing, extending, and modeling, the fact
> that there are models OF T in which G_T is false
> means that referring to G_T as "true"[simpliciter]
> is inappropriate[simpliciter]. Surely the truth-value[s]
> assigned by models OF T are MORE relevant to
> the issue than whatever decision hapens to be
> made by the standard. The standard doesn't become
> relevant until the SECOND incompleteness theorem,
> when we discover that Con(T) is also undecided.
> Since Con(T) *must* be true (not because the standard
> says so, but because that is a necessary condition for
> any of these models to exist at all),
This only makes sense if you avail yourself of a notion of truth
simpliciter, which you just objected to doing.
There is not, as you imply, any good reason to privilege the standard
model in the context of Con(T) but not in the context of G_T (by the
way, G_T and Con(T) are equivalent in Sigma-1-Induction Arithmetic). We
can talk about any model we like. There is no objection to us defining
"true" to mean "true in the standard model", this is justified because
we are quite interested in the notion of truth in the standard model
and want to have a shorthand way of referring to it. On the other hand,
the fact that the truth of Con(T) in the standard model is a necessary
condition for T to have models is no particularly good reason to
"privilege" the standard model. We "privilege" the standard model
because we are interested in it. There can be no other justification,
and none is necessary.
> the fact that the standard
> gets it right while many other models get it wrong THEN
> becomes a reason to privilege the standard. The fact that
> the standard was about&only-about the finite naturals that
> arithmetic was always intended to be about is NOT relevant.
> The fact that sentences and proofs have to be finite is
> part of the reason why the standard's perspective is privileged,
> but the discovery of non-standard models could just as easily
> have invited an expansion of the notions of proof or number to
> accomodate hyper-finite things that were STILL proofs or
> STILL numbers.
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