It's easier to answer that question about the Halting Problem, so here's Scott Aaronson proving it as a corollary to the Halting Problem: http://www.scottaaronson.com/blog/?p=710

Then, defending the value of the Halting Problem as a real problem is easy, relatively generally through Rice's Theorem: http://en.wikipedia.org/wiki/Rice%27s_theorem

In colloquial terms, one can render Rice's Theorem as "Would you like to produce a program that can prove anything remotely interesting across the set of all possible inputs? TOO BAD!" That's glib, but less inaccurate than we'd like.

Godel's theorem is, as Scott said, really sort of ad hoc and hard to easily apply in your head the way you're asking about. The Halting Problem, on the other hand, turns up everywhere, in real life (though many people still often don't realize it).