TITLE: Information-theoretic proof of Gödel's Theorem?
QUESTION [6 upvotes]: 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

REPLY [3 votes]: I think that parts (1) and (2) would already prove the theorem. The heart is part (2), which phrased another way says "no finite number of bits can encode all truths of number theory".   The difficult thing with the proof would be making formal sense of "finite number of bits".