Ordinal

An ordinal is a well ordered set. We identify ordinals that are Order-isomorphic to each-other.

Order Type

Proposition

Let be an ordinal. Then the ordinals strictly less than form a Well-ordered set of Order Type .

Proof

Let be a Well-ordered set whose Order Type is Let is Linear ordered by '' by … The map , sending is an Order-isomorphic Hence, is Well-ordered by and so is

which consists exactly of ordinals .

Theorem

Let be a nonempty set of ordinals. Then has a least element.

Proof

Let . If is not a least element, then By previous, has a least element . Since is an Initial Segment of ordinals (, ) it follows that is a least element of .

Stuff

Burali-Forti paradox Class of Ordinals Supremum of Ordinals Epic list of ordinals Hartogs’ Lemma Types of ordinals Ordinal Addition Ordinal Multiplication Ordinal Exponentiation