Information-theoretic proof of Gödel's Theorem? I'm looking for an information-theoretic proof of G&ouml;del's Theorem that goes something like this, without any reference to diagonalization:


Every axiom system in the scope of G&ouml;del'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
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".
This makes no sense to me.  Yes, some axiom systems can be written down using a finite number of bits.  From those axioms infinitely many theorems may be deduced.  Does it not require infinitely many bits to specify them?  How does your "proof" distinguish between systems that are complete, such as Presburger arithmetic, and systems that are not?