Counting Theorem

Theorem
Every woset is order isomorphic to a unique ordinal.

Existence
Let $\struct {S, \preceq}$ be a woset.

From Condition for Woset to be Isomorphic to Ordinal‎, it is enough to show that for every $a \in S$, the initial segment $S_a$ of $S$ determined by $a$ is order isomorphic to some ordinal.

Let:
 * $E = \set {a \in S: S_a \text{ is not isomorphic to an ordinal} }$

We will show that $E = \O$.

that $E \ne \O$.

Let $a$ be the minimal element of $E$.

This is bound to exist by definition of woset.

So, if $x \prec a$, it follows that $S_x$ is isomorphic to an ordinal.

But for $x \prec a$, we have $S_x = \paren {S_a}_x$ from definition of an ordinal.

So every segment of $S_a$ is isomorphic to an ordinal.

Hence from Condition for Woset to be Isomorphic to Ordinal‎, $S_a$ itself is isomorphic to an ordinal.

This contradicts the supposition that $a \in E$.

Hence $E = \O$ and existence has been proved.

Uniqueness
Uniqueness follows from Isomorphic Ordinals are Equal.

Hence the result.

Also presented as
Some sources use this result as the definition of an ordinal as the order type of a well-ordering.