Re: Sorry Godel - All Truths are Provable

Howdy

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

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

Lemma 1: Correktness of Provability:
|- A => |= A.

Proof: Done by induction on the
proof length usually.

Lemma 2: Completness of Provability:
|= A => |- A.

Proof: Usually shown that if |/- A
then exists M with M |= ~A, i.e.
counter model construction.

Lemma 3:
|= A <=> |- A.

Proof: Follows from Lemma 1 and Lemma 2.

Highly recommended reading:
Ebbinghaus et al.: Mathematical Logic,
Second Edition, Springer Verlag, 1996

IV A Sequent Calculus, 6.2 Theorem on the Correctness
V The Completeness Theorem, 4.1 Completness Theorem

Further References for the Completeness Theorem:
K. Gödel, Die Vollständigkeit (=Completness) der
Axiome des logischen Funktionenkalküls, Monatshefte für
Mathematik und Physik 37 (1930)

L. Henkin, The Completness of First-Order
Functional Calculus, The Journal of Symbolic
Logic 14 (1949)
.

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