Godel's Proof

by Ernest Nagel

Let ‘N’ by definition stand for the class of all normal classes. We ask whether N itself is a normal class. If N is normal, it is a member of itself (for by definition N contains all normal classes); but, in that case, N is non-normal, because by definition a class that contains itself as a member is non-normal. On the other hand, if N is non-normal, it is a member of itself (by definition of “non-normal”); but, in that case, N is normal, because by definition the members of N are normal classes. In short, N is normal if, and only if, N is non-normal. It follows that the statement ‘N is normal’ is both true and false. This fatal contradiction results from an uncritical use of the apparently pellucid notion of “class.”

Russell's Paradox. It blew my mind when I first heard about the barber story back in college days.