One of the four defining axioms of an ultrafilter is that it doesn't contain the empty set (according to Wikipedia, and a talk I was listening to today). Isn't this implied by the other axioms? If an ultrafilter U on X contained the empty set, then it also contains every superset, including X. Therefore, it doesn't contain the complement of X, which is the empty set, and therefore we have a contradiction. I understand that the axiom is necessary if the filter is not ultra to eliminate trivial cases, but am I missing something or is it redundant in the ultra case?