Order on Well-ordered sets

For Well-ordered sets ,
write if is Order-isomorphic to an Initial Segment of

Theorem

Let be Well-ordered sets.
Then either or .

Proof

Assume . Particular
So we can fix
Define by Definition by recursion:

Assume the ‘otherwise’ clause arises. Then there is a least where it arises. So
And for all

As in (previous prop) we show that is order preserving.
So is Order-isomorphic to an Initial Segment of

As in (previous prop) we show is order-preserving and is an initial segment of
Hence .

Proposition

Let , be Well-ordered sets.
If and then is Order-isomorphic to

Proof

Let and be Order-isomorphic to initial segments of and respectively
Then is an order isomorphism to an Initial Segment of .
By Subset Collapse and (prop 3?) .
Similarly

Constructing new well ordered sets

for some and extend the ordering…
Extends
Nested

Proposition

Let be a Nested set of Well-ordered sets. Then there is a well ordered set such that for all .

Proof

Let
For we let iff there is some such that and .
Since the are Nested,
it follows that on is a well defined Linear order
such that each is an Initial Segment of
Let , .
Then there is some such that
Since is Well-ordered,
then has a least element .
Since is initial segment of ,
has to be the least element of .