[lmkg]

If you want to understand just Godel's theorem, you probably want to focus on Logic more than Math or CS. Godel's proof involves qualifying over first-order logic statements and logical properties of numbers much more than it involves computation or numerics. The most nitty-gritty part is Godel encodings, but it's not that computationally intensive, and the details actually aren't that relevant.

If you want to understand the ramifications of Godel's theory, it impacts math more directly than CS. The most directly impacted branches of math are the more "fundamental" ones like Set Theory. Limitations of CS has a lot more to do with Turing's theorems than Godel's.

[IsTom]

Actually Godel's proof is important to CS, it's an analogue of Turing's Proof concerning undecidable problems.

[lcargill99]

My understanding is that Godel's proof was the inspiration for much of Turing's work.

[nooooooo]

The unsolvability of the halting problem has a slightly weaker version of Goedel's first incompleteness theorem as a trivial corollary. You can use Turing machines to prove a stronger statement very easily as well. Therefore the CS perspective is very enlightening. See http://www.scottaaronson.com/blog/?p=710.