Gödel hit upon the rare but powerful property of self-reference. Mathematical versions of self-referring expressions, such as “This statement is not provable in this system,” can be constructed without breaking the rules of mathematical systems. But the so-called self-referring “Gödel statements” introduce contradictions into mathematics: if they are true, then they are unprovable. If they are false, then because they say they are unprovable, they are actually true. True means false, and false means true—a contradiction.
