[mstoehr]

"This was really Godel's point: mathematics is not identical with formalism. They stand and fall separately. This is not to say that mathematics could never collapse for any reason, only that it would take a lot more than finding a paradox at the center of ZFC to make it happen."

That's significant because another great mathematician David Hilbert challenged mathematicians to come up with a complete and consistent set of axioms for all of mathematics. This was a great hope at one time that was shattered by Godel. Most people don't really talk about this old program anymore (except as a historical curiosity) because there is utterly no hope in it at all. The other implications that people try to draw from Godel's work is probably a consequence of the fact that his work sounds like it says so much more than it actually does when it's translated into normal English (and out of math-speak).