Subset Collapse

Let be a Well-ordered set and
Then there is a unique Initial Segment of which is Order-isomorphic to

Proof

Uniqueness

Assume is an order isomorphism from to an initial segment of
By lemma ?

for all
By Proof by Induction, is uniquely determined

Existence

WLOG
Fix
Define by recursion

We first prove that the ‘otherwise’ clause does not arise
We do this by proving that for all
(by induction)
Fix and assume for all
Then and hence

Fix in

Hence and so so is order preserving

Let
We show that for all
It will then follow that is an Initial Segment of and we are done.
Proof by Induction
Fix and assume for all

so and thus