Personally I find Godel incompleteness to be a far more disturbing discovery than any of the things listed in the article. Something about a fundamental limitation on deductive systems just scares the hell
out of me.
The idea that there is a fundamental limit to how far you can get when starting from something declared true by fiat has always struck me as kind of obvious. Though that's probably at least partly because I grew up well after the invention of science and the work of people like Godel.
Godel's theorem implies that in a consistent system there are unprovable truths. I've always wondered if those truths could be things no one would care about anyway. Are statements like "this statement is unprovable" useful for anything other than proving the Incompleteness theorem?
The halting problem seems kind of important. Also, at least two of Hilbert's problems -- the continuum hypothesis and solving general Diophantine equations -- turned out to be undecidable (the latter because it reduces to the halting problem.)
Edit: Er, actually, the Diophantine equation question would be uncomputable (like the halting problem), not undecidable. The ideas are closely related, though.
