Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.
It is often argued (you know, by nerds) that this theorem proves it is impossible for us to ever fully understand the world, since understanding it would imply we have a theory of it—and surely any theory about the world must include “basic arithmetic truths”!
These naysayers overlook one crucial loophole: Gödel’s theorem does not imply that our knowledge must be incomplete—only that it cannot be both complete and consistent. What if it’s inconsistent?
Maybe reality is inconsistent with itself. If so, I don’t believe anyone’s proven we are doomed to fail in striving to understand it. Ever think about that?
Ha! I didn’t think so.