# Properties of Class of All Ordinals/Zero is Ordinal

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## Theorem

Let $\On$ denote the class of all ordinals.

The natural number $0$ is an element of $\On$.

## Proof

We have the result that Class of All Ordinals is Minimally Superinductive over Successor Mapping.

Hence $\On$ is *a fortiori* a superinductive class with respect to the successor mapping.

Hence, by definition of superinductive class:

- $\O \in \On$

We identify the natural number $0$ via the von Neumann construction of the natural numbers as:

- $0 := \O$

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 $1.2 \ (1)$