Counting Theorem

Theorem
Every woset is order isomorphic to a unique ordinal.

Proof

 * First we prove existence.

Let $$\left({S; \preceq}\right)$$ be a woset.

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

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

We will show that $$E = \varnothing$$.

So, suppose that $$E \ne \varnothing$$.

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 $$S_x = \left({S_a}\right)_x$$ 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 $$E = \varnothing$$ and existence has been proved.


 * Uniqueness follows from Isomorphic Ordinals are Equal.

Hence the result.