 One question that has been important from the beginning of set theory concerns the difference between sets and proper classes. (This question has a natural counterpart for category theory: the difference between small and large categories.) Cantor’s diagonal argument forces us to recognize that the set-theoretical universe as a whole cannot be regarded as a set. Cantor’s Theorem shows that the power set (i.e., the set of all subsets) of any given set has a larger cardinality than the given set itself. Now suppose that the set-theoretical universe forms a set: the set of all sets. Then the power set of the set of all sets would have to be a subset of the set of all sets. This would contradict the fact that the power set of the set of all sets would have a larger cardinality than the set of all sets. So we must conclude that the set-theoretical universe cannot form a set.