I always read the second incompleteness theorem as important /because/ of the connection to truth. I think we (many, anyway) want to believe something like Lei

I always read the second incompleteness theorem as important /because/ of the connection to truth. I think we (many, anyway) want to believe something like Leibniz' principle of sufficient reason: that every true statement has some sufficient reason for being so. We naturally interpret proofs in a formal system as that "sufficient reason". If we could prove from within a system that all and only consistent statements have a proof -- then that's great news! Maybe that system is Truth, or if it isn't maybe that's evidence some other system like it could be. So Godel's work crushed this hope. No system can prove itself consistent, so there is no system under which the principle of sufficient reason is true. Maybe then we shouldn't believe it! Then are there truths that are true for no reason at all...? But of course, there are many readings here -- many different roads to being thwarted by Godel :)