Well-ordered

A well-ordering of a set is a Linear order on
s.t. every nonempty subset of has a least element:

This least element is unique by antisymmetry of

This property is preserved by Order-isomorphic.

Lemma

Let , be well-ordered sets
Let be an Initial Segment of
Let be an Order-isomorphic.
Then for every , we have

Corollary

Proof by Induction

Proposition

Let , be well ordered sets that are Order-isomorphic.
Then there is a unique order-isomorphism

Proof

Assume are order-isomorphisms.
We prove by Proof by Induction
Fix .
Assume (induction hypothesis)
By the lemma, where
and where
By induction hypothesis, so
By Proof by Induction,