Cardinal Number is Ordinal

Theorem

Let $S$ be a set such that $S \sim x$ for some ordinal $x$.

Let $\card S$ denote the cardinality of $S$.

Then:

$\card S \in \On$

where $\On$ denotes the class of all ordinals.

Proof

If $S \sim x$, then $\set {x \in \On: S \sim x}$ is a non-empty set of ordinals.

It follows that this set has a minimal element, its intersection.

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This minimal element is the cardinal number of $S$, by the definition of cardinal number.

Thus, it is an ordinal.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.8$