# Well-Ordering of Class of All Ordinals under Subset Relation

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## Theorem

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

$\On$ is well-ordered by the subset relation such that the following $3$ conditions hold:

\((1)\) | $:$ | the smallest ordinal is $0$ | ||||||

\((2)\) | $:$ | for $\alpha \in \On$, the immediate successor of $\alpha$ is its successor set $\alpha^+$ | ||||||

\((3)\) | $:$ | every limit ordinal is the union of the set of smaller ordinals. |

## Proof

We have that Class of All Ordinals is $g$-Tower.

By Zero is Smallest Ordinal, $0$ is the smallest element of $\On$.

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

- $0 := \O$

The result then follows directly from $g$-Tower is Well-Ordered under Subset Relation.

$\blacksquare$

## 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 $1.11$