TITLE: Ultra Filter and Axiom of Choice
QUESTION [8 upvotes]: Some person said me: "The fact that Ultra Filters exist is equivalent to the Axiom of choice".
Is this correct? I nees some good references about the subject, please help me.
Thanks

REPLY [4 votes]: It is not true. First, there are always principal (or fixed) ultrafilters: if $S$ is any non-empty set, and $s\in S$, $\{U\subseteq S:s\in U\}$ is a principal (or fixed) ultrafilter on $S$. Note that in this case $\bigcap\mathscr{U}=\{s\}$.
An ultrafilter $\mathscr{U}$ on a set $S$ is free if $\bigcap\mathscr{U}=\varnothing$. There are no free ultrafilters on any finite set. The existence of free ultrafilters on infinite sets requires some amount of choice, but less than the full axiom of choice. Specifically, the assertion that every filter can be extended to an ultrafilter is equivalent to the Boolean prime ideal theorem, which is independent of ZF but, by a result of Halpern and Levy, strictly weaker than the axiom of choice.