One proves the existence of nonprincipal ultrafilters with the axiom of choice. (You can show “every filter is contained in some ultrafilter” pretty directly using Zorn’s lemma.) You don’t actually need the full strength of AC—I think “every infinite set has a nonprincipal ultrafilter” is weaker than countable choice, for instance—but you definitely need some choice, and in particular “there are no nonprincipal ultrafilters” is known to be consistent with ZF.
Thanks for the advice! Still learning how to phrase things correctly & effectively.
I wasn’t aware that you can’t actually explicitly construct a nonprincipal ultrafilter—this is interesting and nonintuitive to me!
One proves the existence of nonprincipal ultrafilters with the axiom of choice. (You can show “every filter is contained in some ultrafilter” pretty directly using Zorn’s lemma.) You don’t actually need the full strength of AC—I think “every infinite set has a nonprincipal ultrafilter” is weaker than countable choice, for instance—but you definitely need some choice, and in particular “there are no nonprincipal ultrafilters” is known to be consistent with ZF.