Every filter on a Boolean algebra (or more generally, any subset with the finite intersection property) is contained in an ultrafilter (see ultrafilter lemma) and free ultrafilters therefore exist, but the proofs involve the axiom of choice (AC) in the form of Zorn's lemma. On the other hand, the statement that every filter is contained in an ultrafilter does not imply AC. Indeed, it is equivalent to the Boolean prime ideal theorem (BPIT), a well-known intermediate point between the axioms of Zermelo–Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC). In general, proofs involving the axiom of choice do not produce explicit examples of free ultrafilters, though it is possible to find explicit examples in some models of ZFC; for example, Gödel showed that this can be done in the constructible universe where one can write down an explicit global choice function. In ZF without the axiom of choice, it is possible that every ultrafilter is principal.
Every filter on a Boolean algebra (or more generally any subset with the finite intersection property) is contained in an ultrafilter (see ultrafilter lemma) and free ultrafilters therefore exist but the proofs involve the axiom of choice (AC) in the form of Zorn's lemma On the other hand the statement that every filter is contained in an ultrafilter does not imply AC Indeed it is equivalent to the Boolean prime ideal theorem (BPIT) a well-known intermediate point between the axioms of Zermelo–Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC) In general proofs involving the axiom of choice do not produce explicit examples of free ultrafilters though it is possible to find explicit examples in some models of ZFC; for example Gödel showed that this can be done in the constructible universe where one can write down an explicit global choice function In ZF without the axiom of choice it is possible that every ultrafilter is principal