Cardinal of Finite Ordinal
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In particular: Predecessor of Finite Ordinal is Finite Ordinal/Element of Finite Ordinal is Finite Ordinal.
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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.
In particular: Predecessor of Finite Ordinal is Finite Ordinal/Element of Finite Ordinal is Finite Ordinal.
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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.
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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: $\S 10.19$
