“This statement is not provable.” Think about it for a minute. It’s a strange statement, since it talks about itself—in fact, it asserts that it is not provable. Let’s call this statement “Statement A.” Now, suppose Statement A could indeed be proved. But then it would be false (since it states that it cannot be proved). That would mean a false statement could be proved—arithmetic would be inconsistent. Okay, let’s assume the opposite, that Statement A cannot be proved. That would mean that Statement A is true (because it asserts that it cannot be proved), but then there is a true statement that cannot be proved—arithmetic would be incomplete. Ergo, arithmetic is either inconsistent or incomplete.”