I remember a joke from college that went something like this: There was this mathematics professor that used to tell his students that if they failed at proving a conjecture they should try to prove the opposite, and if they couldn't do that either they should quit mathematics. He had to stop saying that after Gödel published his paper. :)

I haven't studied Gödel's proof and have to admit I would probably have to brush up quite a bit on my maths to be able to, but to me this simple joke offers a more pedagogical and perhaps more meaningful understanding of Gödel's (first) incompleteness theorem.