>And btw, has anyone else felt that the Halting Problem and Godel's theorem are two sides of the same coin? Is there any formal connection between them? I feel like they are not "independent" (in the sense that Euclid's 5th postulate is independent).

The theorem of the unsolvability of the halting problem is used in modern proofs of Gödels incompletness teorem(s) [1]. Hofstadter writes about this relation in Gödel Escher Bach as well.

[1] http://www.amazon.com/Lectures-Logic-Set-Theory-Mathematical...