I'm looking for an information-theoretic proof of Gödel's Theorem that goes something like this, without any reference to diagonalization:
- Every axiom system in the scope of Gödel's Theorem has a finite number of bits.
- It requires an infinite number of bits to specify the all the truths of number theory.
- By the Soundness theorem, no new bits can be introduced by deduction.
- So no such axiom system as specified in part 1 above can fully axiomatize number theory.
Does such a proof exist? Is it even feasible? Please include references with your answer. Thanks