Re: The incompleteness theorems, Sigma-1-completeness, induction, all that


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), 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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