EGB is one of my favourite books ever, but I think it doesn't explain Gödel's theorem as simple as it could. Of course, there are just so many wonderful things in EGB that this is not a fatal flaw by any means.

There is a fantastic book called "The Universal Computer: The Road from Leibniz to Turing" by Martin D. Davis that makes a very good job of explaining the context of Gödel's theorem, the theorem itself and the relation with computing in an accesible way. I'd recommend it to anyone interested either in computers or in Gödel's theorem.