Ordinal number

Set

 definiendum $\alpha\in \mathrm{Ord}$ inclusion $\alpha$…transitive for all $\beta,\gamma\in\alpha$ postulate $(\beta\in\gamma)\ \lor\ (\gamma\in\beta)\ \lor\ (\beta=\gamma)$

Discussion

The second requiement says that the ordinal admits a set theoretical constuction of a certain order relation for all its elements. The first requirement means $\ \forall (\beta \in \alpha).\ \beta \subseteq \alpha\$ and both together imply that ordinals represent stackings of other ordinals.

$\mathrm{Ord}$ is not a set, but a proper class.

Predicates

For any two ordinals $\in$ gives an ordering $<$ via

 predicate $\beta<\gamma\equiv \beta\in\gamma$

Reference

Wikipedia: Ordinal number