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