\begin{definition}[Definition:Minimal Infinite Successor Set/Definition 1]
Let $S$ be an infinite successor set.
The '''minimal infinite successor set''' $\omega$ is the infinite successor set given by:
:$\ds \omega := \bigcap \set {S' \subseteq S: S' \text{ is an infinite successor set} }$
that is, $\omega$ is the intersection of every infinite successor set which is a subset of $S$.
\end{definition}