There does not exist a set of all sets

Let U = the universe and suppose U were such a set. Then if A is any set that is an element of U. The property P(A): A is not an element of A is a meaningful property for elements of U. Let M = {A; A is an element of U and A is not an element of A}. By the axiom of specification, M is a set; and therefore an element of U. By the law of the excluded middle M is an element of M or M is not an element of M.

But if M is an element of M then, by definition, M is an element of U and M is not an element of M so that M is an element of M and M is not an element of M which is a contradiction.

Alternately if M is not en element of M, then this is so only if M is not an element of U or M is an element of M. But M is an element of U so that we must have M is an element of M. Thus M is not en element of M and M is an element of M, again a contradiction.

From Elements of Set Theory 2nd edition by Zehna and Johnson, 1972

In either case it is a contradiction so that U does not exist