Model here, model there

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?

.

Relevant Pages

  • Re: Model here, model there
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  • Re: Godel Contradiction
    ... incomplete" but rather "IF theory T is consistent then theory T is ... true (recall 'true' means 'true in the standard model of PA' in this ... and how do you prove your meta-theory to be consistent. ... meta-theory, ad infinitum, up through meta-meta...theories. ...
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  • Re: Model here, model there
    ... then contains an undecidable sentence G. ... If T is consistent then G is ... ordinary mathematical reasoning" means. ... G is true in the standard model ...
    (sci.logic)
  • Re: Model here, model there
    ... then contains an undecidable sentence G. ... If T is consistent then G is ... ordinary mathematical reasoning" means. ... G is true in the standard model ...
    (sci.logic)
  • Re: Whats up with gravity wave detection?
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