Any ideas on how to do this one?
Let A be a set. Show that a complement of A does not exist. So I need to show that there isn't a set of all x not in A.
is a set and ?
Assuming axioms of ZF set theory, can't be a set: If were a set, then since is a set, axiom of pairing gives that is a set. Axiom of union then tells us that that there exist a set whose elements are elements of and elements of . But from the definition of we get that is the universal class of all sets, which is a proper class, i.e. is not a set. This is a contradiction, thus can't be a set.
Can you see why the class of all sets is not a set?