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,