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 .