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:

$\left|{ \left({ \left|{S}\right| }\right) }\right| = \left|{S}\right|$

where $\left|{S}\right|$ denotes the cardinal number of $S$.


Corollary

Let $\mathcal N$ denote the class of all cardinal numbers.

Let $x$ be an ordinal.


Then:

$x \in \mathcal N \iff x = \left|{ x }\right|$


Proof

By Condition for Set Equivalent to Cardinal Number:

$S \sim \left|{S}\right|$

Therefore, by Equivalent Sets have Equal Cardinal Numbers:

$\left|{S}\right| = \left|{ \left({ \left|{S}\right| }\right) }\right|$

$\blacksquare$


Sources