\begin{definition}[Definition:Principal Ultrafilter/Nonprincipal]
Let $S$ be a set.
Let $\powerset S$ denote the power set of $S$.
Let $\FF \subset \powerset S$ be an ultrafilter on $S$ which does not have a cluster point.
Then $\FF$ is a '''nonprincipal ultrafilter'''  on $S$.
\end{definition}