# Natural Number is Ordinal

## Theorem

Let $n \in \N$ be a natural number.

Then $n$ is an ordinal.

## Proof 1

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.

$\blacksquare$

## Proof 2

From the von Neumann construction of the natural numbers, $\N$ is identified with the minimally inductive set $\omega$.

From Superinductive Class under Successor Mapping contains All Ordinals, it follows by the Principle of Mathematical Induction that every natural number is an ordinal.

$\blacksquare$

## 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 $1.5$