Distinct Ordinals are not Order Isomorphic

Theorem

Let $\alpha$ and $\beta$ be ordinals such that $\alpha \ne \beta$.

Then $\alpha$ and $\beta$ are not order isomorphic.

Proof

By definition, an ordinal is well-ordered by the subset relation.

From Class of All Ordinals is Well-Ordered by Subset Relation, the class of all ordinals is a nest.

Hence:

$\paren {\alpha \subsetneqq \beta} \lor \paren {\beta \subsetneqq \alpha}$

This needs considerable tedious hard slog to complete it. In particular: there should be a result somewhere that explicitly states that one ordinal is an initial segment of another one 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.

The result follows by Well-Ordered Class is not Isomorphic to Initial Segment.

$\blacksquare$

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 $2.5$