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