Re: Model here, model there
- From: LauLuna <laureanoluna@xxxxxxxx>
- Date: Thu, 25 Oct 2007 12:23:02 -0700
On Oct 25, 4:50 pm, 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. No matter how long we generate
theorems we will never derive a contradiction. If this is the case
then Con(T) is plainly true. We can prove G = 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)
So is the assumption that T is consistent equivalent to: the standard
model is true and the non-standard models are false?
As Peter Smith and George have already pointed out, models can't be
true or false.
But there is another problem. It looks like you are conflating formal
sentences with propositions. You can't say 'Con(T) is plainly true',
because Con(T) is just a formula that is only true (or false) under
some interpretation. Formal sentences only express propositions by
means of interpretations. And only propositions can have a truth
value.
That T is consistent is plainly true. Con(T) is true in the standard
model and in some nonstandard ones, for consider that T+Con(T) still
yields an incomplete system with divergent models.
Regards
.
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