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