# Cardinal of Finite Ordinal

## Theorem

Let $n$ be a finite ordinal.

Let $\left|{n}\right|$ denote the cardinal number of $n$.

Then:

- $\left|{n}\right| = n$

## Proof

Since $n$ is an ordinal, it follows that $\left|{n}\right| \le n$ by Cardinal Number Less than Ordinal: Corollary.

Hence, $\left|{n}\right|$ is also a finite ordinal.

Since $n$ is an ordinal, it also follows that $n \sim \left|{n}\right|$ by Ordinal Number Equivalent to Cardinal Number.

By Equality of Natural Numbers, it follows that $n = \left|{n}\right|$.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 10.19$