\begin{definition}[Definition:Ordinal/Definition 1]
Let $S$ be a set.
Let $\Epsilon \! \restriction_S$ be the restriction of the epsilon relation on $S$.
Then $S$ is an '''ordinal''' {{iff}}:
:$S$ is a transitive set
:$\Epsilon \! \restriction_S$ strictly well-orders $S$.
\end{definition}