| definiendum | α∈Ord |
| inclusion | α…transitive |
| for all | β,γ∈α |
| postulate | (β∈γ) ∨ (γ∈β) ∨ (β=γ) |
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 ∀(β∈α). β⊆α and both together imply that ordinals represent stackings of other ordinals.
Ord is not a set, but a proper class.
For any two ordinals ∈ gives an ordering < via
| predicate | β<γ≡β∈γ |
Wikipedia: Ordinal number