# Non-principal ultrafilters on a set [duplicate]

Let $X$ be a set. If $X$ is finite then all ultrafilters on $X$ are principal, i.e. have the form $\{A \subseteq X : x \in A\}$ for some $x\in X$.

But now suppose $X$ is infinite, say $X=\mathbb N$. Is there any concrete example of a non-principal ultrafilter on $X$? And does one need the axiom of choice to prove their existence?

## marked as duplicate by Kyle, Community♦May 4 '16 at 17:10

We know this because there are models of $\sf ZF$ where every ultrafilter over the natural numbers is principal. In fact there are models where every ultrafilter over any set is principal.