Class of Infinite Cardinals is Proper Class

Theorem

The class of infinite cardinals $\NN’$ is a proper class.

Proof

Aiming for a contradiction, suppose $\NN'$ is a small class.

By Union of Small Classes is Small, $\NN’ \cup \omega$ is a small class.

By definition of the class of infinite cardinals, $\NN \subseteq \NN' \cup \omega$.

But by Axiom of Subsets Equivalents, this means that $\NN$ is a small class.

This contradicts Class of All Cardinals is Proper Class.

Therefore, $\NN’$ is not a small class.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.43$