Re: Model here, model there
- From: Newberry <newberryxy@xxxxxxxxx>
- Date: Thu, 25 Oct 2007 19:38:55 -0700
On Oct 25, 8:21 am, Peter_Smith <ps...@xxxxxxxxx> wrote:
On 25 Oct, 15:50, Newberry <newberr...@xxxxxxxxx> wrote:
Let T be a theory consisting of classical logic with Peano axioms. T
then contains an undecidable sentence G. If T is consistent then G is
intuitively true, it is true on interpretation, and it is true in the
standard model. They tell me they are pretty sure that T "is"
consistent, i.e. Con(T). We cannot prove it but it is consistent by
ordinary mathematical reasoning.
I'm not quite sure what "we cannot prove it but it is consistent by
ordinary mathematical reasoning" means. If ordinary mathematical
reasoning involves at least the principles of Zermelo set theory
(forget about the Fraenkel bit and Choice!) then that's enough to
prove Con(T).
So assume T is consistent. Then we have.
1) Con(T) is plainly true
2) G is true in the standard model
3) G = Con(T)
Is there an implication there that you need to assume T is consistent
to prove the equivalence of G and Con(T)? You don't.
No. I do not see where I implied that.
So is the assumption that T is consistent equivalent to: the standard
model is true and the non-standard models are false?
Eh? It makes no sense so say that the standard model is true. It is
sentences which are are true or false, not models.
Does itv mean then that T is consistent in the standard model or that
G is plainly true?
.
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