Well-Ordering of Class of All Ordinals under Subset Relation

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 $1.11$