\begin{definition}[Definition:Infinite Successor Set]
{{mergeto|Definition:Inductive Set}}
Let $S$ be a set.
Then $S$ is an '''infinite successor set''' {{iff}}:
:$\O \in S$
:$x \in S \implies x^+ \in S$
where $\O$ is the empty set, and $x^+$ denotes the successor set of $x$.
\end{definition}