Gottlob Frege said (in essence) that any set either:
a) Contains itself
b) Doesn't contain itself
Makes sense, right?
Later on, Bertrand Russell pointed out this interesting paradox:
Let A be a set containing all sets that don't contain themself. Now, as yourself, should A be a member of itself? It doesn't contain itself, so it should, right? Now, it does contain itself, so it can't be in there. That's the paradox!
Incidentally, I guess after Russell pointed this out to Frege, he came close to committing suicide. It goes to show: inventing axioms is serious business. :|
Heh, I remember learning about an internal contradiction in set theory one time. That may have been it :)
Cool.. :)
Haha, that's so awesome.