Is a non-principal ultrafilter the same thing as a free ultrafilter?

Question

Can someone please confirm if a non-principal ultrafilter is the same thing as a free ultrafilter. I keep finding conflicting definitions so am not sure.

Answer 1

They're the same. Sometimes "free" is simply defined to mean "nonprincipal". Another definition given is:

An ultrafilter $\mathcal{F}$ is free if $\bigcap \mathcal{F} = \emptyset$.

But that definition is equivalent to "nonprincipal":

• Clearly, a free ultrafilter is nonprincipal (even more clearly, a principal ultrafilter is not free).
• If an ultrafilter $\mathcal{F}$ on a set $X$ is nonprincipal, it contains $C_x \colon = X \setminus \{x\}$ for every $x \in X$, so its intersection is contained in the intersection of all of $C_x, x \in X$, which is $\emptyset$.

(If there's yet another definition of "free", I've never encountered it.)