Re: Provable in T?

Rupert wrote:
If T is consistent, then G_T will be true in the standard model and
also provable in T+Con(T).

"T" has been murderoulsy ambiguous throughout this whole
discussion. It does actually matter what T is.
From the fact that it starts with T, one might've thought
it was safe to assume that T was a theory. Usage in
the discussion belies that, unfortunately. The standard
model has a theory, and if you want to concentrate on the
standard model, then there is a sense in which it doesn't matter
whether you use the model or the theory; one poster, in a related
thread, correctly observed that models were analogues of complete
theories. AK ignorantly disagreed with this for reasons that remain
unclear; I hope they didn't have to do with thinking about
completeness-
as-covering-truths as opposed to negation-completeness (which is
the ONLY important kind for this discussion, REGARDLESS of Hilbert's
original objectives).

In the original Godel proof, T was PM.
For modern purposes, the initial basic T needs to be PA.
The Incompleteness theorems invite the extension of PA
(or whatever T they were being applied to; for concreteness,
again, we will start with PA) by new axioms, axioms which they
prove to be all of new, axiomatic, and needed, by proving that
the original T, being incomplete, did not decide them.

Rupert continued,

However, T+Con(T) may not be consistent.

There were so many other things to object to in this message
that I never got around to focusing on the rebuttal of this one.
But Rupert has explained this incorrectly.
T + Con(T) IS ABSOLUTELY guaranteed to be consistent,
AS is T ~ Con(T). The only question (given the particular construction
of G_T in G1) is which of them G_T will be equivalent to. If it is
equivalent
at all. Obviously, having the same truth value in ONE model but NOT
in others will PRECLUDE (DEFINITIONALLY!) any EQUIVALENCE.

In the original reply, I did in fact say (though we never got around
to
discussing it until now, because I said it last and there was too much
intervening pollution from other issues),

The whole import of G2 is that T+Con(T) and T~Con(T) are
BOTH consistent if T is.

.

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