Natural Number is Ordinal/Proof 1
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Theorem
Let $n \in \N$ be a natural number.
Then $n$ is an ordinal.
Proof
Consider the class of all ordinals $\On$.
From Class of All Ordinals is Minimally Superinductive over Successor Mapping, $\On$ is superinductive.
Hence a fortiori $\On$ is inductive.
From the von Neumann construction of the natural numbers, $\N$ is identified with the minimally inductive set $\omega$.
By definition of minimally inductive set:
- $\omega \subseteq \On$
and the result follows.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers: Theorem $1.5$