\begin{definition}[Definition:Ordering on Natural Numbers/Von Neumann Construction]
Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction.
The '''strict ordering''' of $\omega$ is the relation $<$ defined by:
:$\forall m, n \in \omega: m < n \iff m \subsetneq n$
The '''(weak) ordering''' of $\omega$ is the relation $\le$ defined by:
:$\forall m, n \in \omega: m \le n \iff m \subseteq n$
\end{definition}