Cardinal of Finite Ordinal

Theorem

Let $n$ be a finite ordinal.

Let $\card n$ denote the cardinal number of $n$.

Then:

$\card n = n$

Proof

Since $n$ is an ordinal, it follows that $\card n \le n$ by Cardinal Number Less than Ordinal: Corollary.

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Hence, $\card n$ is also a finite ordinal.

Since $n$ is an ordinal, it also follows that $n \sim \card n$ by Ordinal Number Equivalent to Cardinal Number.

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By Equality of Natural Numbers, it follows that $n = \card n$.

$\blacksquare$

Sources

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