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?(adsbygoogle = window.adsbygoogle || []).push({});

If an ultrafilter U on X contained the empty set, then it also contains every superset, including X. Therefore, itdoesn'tcontain 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?

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# Empty set not in an ultrafilter

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