The treatment of Godel's theorems in GEB always struck me as gratuitous: It treats a technical result as having great philosophical import. The philosophical content of Godel's theorems is already found in the Liar paradox, "This statement is false." and incompleteness in a grand sweeping sense involving language in its entirety can be found in works as early as the Daoists and the Greeks.
Edit: I should say that this wasn't at all obvious to me when I was in college and thinking hard about incompleteness. The limitations of mathematics disturbed me deeply. These days, I think it's part of the beauty of mathematics that its limitations are its own result.
> The philosophical content of Godel's theorems is already found in the Liar paradox, "This statement is false."
Sort of. Common reactions to the English version of the liar paradox are along the lines of 'that's meaningless so it doesn't imply anything' or 'so natural language is inconsistent, what do you expect?' Most of the work Gödel did was to find a way to encode meta-logic into formal language in such a way that he could then state the liar paradox in a format so ironclad that it could no longer be dismissed.
Sure, that's the technical content of Godel :) If people chose not to take the Liar paradox seriously, I think they would gnash their teeth just as vigorously at the incompleteness theorem, and there is an expansive graveyard of attempts to circumvent it.
In my opinion the most important bit of Gödel's results is Tarski's Undefinability, an essential corollary that is often completely omitted when talking about incompleteness. GEB covers this (on p580 in my copy) and talks a bit about the consequences.
The other important part, of course, is that these results are inescapable.
Edit: I should say that this wasn't at all obvious to me when I was in college and thinking hard about incompleteness. The limitations of mathematics disturbed me deeply. These days, I think it's part of the beauty of mathematics that its limitations are its own result.