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With the axiom of choice, we are left with two options for ultrafilters:

a) principal ultrafilters, built from a singleton {x}.

b) nonpricipal filters of which all contain the cofinite filter, ergo complements of finite sets subalgebras.

Isn't this kind of flimsy? To get to more exotic/exciting objects does one:

give up AC

or

gives up the ultra in ultafilter (the A or X\A is in F condition)?