===== Ordinal number =====
==== Set ====
| @#FFBB00: definiendum | @#FFBB00: $\alpha\in \mathrm{Ord}$ |
| @#AAFFAA: inclusion   | @#AAFFAA: $\alpha$...transitive |
| @#FFFDDD: for all     | @#FFFDDD: $\beta,\gamma\in\alpha$ |
| @#55EE55: postulate   | @#55EE55: $ (\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

| @#EEEE55: predicate   | @#EEEE55: $\beta<\gamma\equiv \beta\in\gamma$ |

=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Ordinal_number|Ordinal number]]
==== Parents ====
=== Requirements ===
[[Transitive relation]]
=== Related ===
[[Trichotomous relation]]