Counting Theorem/Corollary

Corollary to Counting Theorem

Every properly well-ordered proper class is order isomorphic to the class of all ordinals.

Proof

Let $A$ be a properly well-ordered class.

Let $\On$ denote the class of all ordinals.

By the Axiom of Replacement, neither $A$ nor $\On$ can be order isomorphic to a proper lower section of the other.

Hence it must be that $A$ is order isomorphic to $\On$.

$\blacksquare$

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: $\S 4$ The counting theorem: Theorem $4.1$ (The counting theorem)