Cardinal of Cardinal Equal to Cardinal

Theorem

Let $S$ be a set such that $S$ is equivalent to its cardinal.

If the axiom of choice holds, then this condition holds for any set.

Then:

$\card {\paren {\card S} } = \card S$

where $\card S$ denotes the cardinal number of $S$.

Corollary

Let $\NN$ denote the class of all cardinal numbers.

Let $x$ be an ordinal.

Then:

$x \in \NN \iff x = \card x$

Proof

By Condition for Set Equivalent to Cardinal Number:

$S \sim \card S$

Therefore, by Equivalent Sets have Equal Cardinal Numbers:

$\card S = \card {\paren {\card S} }$

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.16$