Class of All Cardinals is Proper Class

Theorem

The class $\NN$ of all cardinal numbers is a proper class.

Proof

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

By Class of All Cardinals is Subclass of Class of All Ordinals:

$\NN \subseteq \On$

Therefore, $\bigcup \NN$ is an ordinal by Union of Set of Ordinals is Ordinal.

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Take $x = \set {y \in \On: y \preccurlyeq \bigcup \NN}$.

By Cardinal Equal to Collection of All Dominated Ordinals, $x$ is a cardinal number and there is no injection $f: x \to \bigcup \NN$.

But since $x$ is a cardinal number, $x \in \NN$, so $x \subseteq \bigcup \NN$ by Set is Subset of Union.

Therefore, the identity mapping $I_x: x \to \bigcup \NN$ is an injection.

This is a contradiction.

Therefore by Proof by Contradiction $\NN$ is not a small class.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.41$