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Best summation of Gödel and paradoxes I think I've ever read:
Godel proved that any consistent mathematical system that is powerful enough to prove anything interesting will include true statements that can't be proven via that system. So from the standpoint of any logic, there will be elements that must be taken "on faith"A healthy way to think about proof systems (where "proof systems" is intended to stand in for a host of mathematic and scientific concepts as well) is that though they are imperfect in the Godel way, they are the best tool that we have for understanding the world and they get great tangible results in terms of science and technology in a way that nothing else does.
