Re: Sorry Godel - All Truths are Provable

Jan Burse wrote:

XX |- PR('A') <=> XX |= A. (2)

The original question was whether (2) can hold. And my claim is yes, (2) can hold.

Any problems with that?

Rupert wrote to Charlie-Boo

What logic? I don't understand your reasons for thinking the
truth-value of a statement has to coincide with its provability-value.

to which you answered

Doesn't have to, but for most logics it does, respectively one can
develop valuations that coincide with some deductive systems.

and referred to the completeness theorem for first order logic.

It seems that your last reply amounts to noting that for certain theories - e.g. 1-consistent extensions of Q - we have that

T |- Prov_T('A') <=> T |= A

This observation, correct as it is, is not any more relevant to Rupert's puzzlement than the completeness theorem in its usual form, and I don't see how it could sensible be construed as the "truth-value" of a statement coinciding with its "provability-value". (2) just says that A is provably-in-T provable in T just in case it is true in all models of T. Consider A = "T is consistent"; A is certainly true, but it is not provable in T, even though we still have T |- Prov_T('A') <=> T |= A.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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