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 i

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.