Two plus two is four. You can prove that two plus two is four. You can prove that you can prove that two plus two is four. And you can prove that you can prove that you can prove that two plus two is four, and so on.
Two plus two is not five. You can prove that two plus two is not five. You can't prove that two plus two is five, or else math is a lot of bunk. But, if math is not a lot of bunk, you can't prove that you can't prove that two plus two is five.
(Shortened from Gödel's Second Incompleteness Theorem Explained in Words of One Syllable by George Boolos, Mind, Vol. 103, January 1994, pp. 1 - 3.)
