Distinct Lower Sections of Well-Ordered Class are not Order Isomorphic

Theorem

Let $\struct {A, \preccurlyeq}$ be a well-ordered class.

Let $L_1$ and $L_2$ be distinct lower sections of $\struct {A, \preccurlyeq}$.

Then $L_1$ and $L_2$ are not order isomorphic with respect to $\preccurlyeq$.

Proof

A lower section of $A$ is a subclass of $A$.

Hence by definition of well-ordered class. $L_1$ and $L_2$ are themselves well-ordered classes.

We have $L_1 \ne L_2$.

The result follows from

This needs considerable tedious hard slog to complete it. In particular: Result needed here to the effect that distinct well-ordered classes are not isomorphic. We have this for sets but not classes. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 2$ Isomorphisms of well orderings: Corollary $2.6$