Re: Which theories hold a true but unprovable statement?

On Feb 8, 12:55 am, jackbat...@xxxxxxxxx wrote:
Hi,
Godel's first incompleteness theorem states that any consistent theory
capable of expressing basic aritmetical truths about numbers is
incomplete, i.e. there exists a wff S such that neither S nor not(S)
is provable in the theory. But the most interesting thing is that S is
true, model-wise. So we "know" that there exists a theorem that cannot
be proven.

My questions are:

- Does the same condition hold to other first-order theories? I.e. to
have an undecidable but true (model-wise) wff. (Obviously I know that
if so, that's not proven via Godel's theorem, but via other
theorems... do you know which ones?)

- Are these theories consistent anyway?

Thanks.

Before you can make sense of the idea of a sentence being true, you
have to specify a model for the theory. Consider the theory of real-
closed fields, for example. This theory is recursively axiomatizable,
and a sentence is true in the model consisting of the real numbers if
and only if it is provable in the theory.

.

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