Set of Natural Numbers is Ordinal
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Theorem
The set of natural numbers $\N$ is an ordinal.
Proof
From Natural Number is Ordinal, every element of $\N$ is an ordinal.
From Union of Set of Ordinals is Ordinal, $\bigcup \N$ is therefore itself an ordinal.
From Set of Natural Numbers Equals its Union:
- $\bigcup \N = \N$
Hence the result.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 3$ Some ordinals