Cardinal of Cardinal Equal to Cardinal

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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