| 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) $ |
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.
For any two ordinals $\in$ gives an ordering $<$ via
| predicate | $\beta<\gamma\equiv \beta\in\gamma$ |
Wikipedia: Ordinal number