## 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