Counting Theorem/Motivation

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Motivation for the Counting Theorem

What we have achieved with the Counting Theorem is that for any properly well-ordered collection $A$, we can assign ordinal numbers as indices of the elements of $A$, treating the latter as a sequence:

the $1$st, the $2$nd, $\ldots$, the $\alpha$th, $\ldots$ elements of $A$

and moreover, we can assign these indices in an order-preserving way.

That is, for all ordinal numbers $\alpha$ and $\beta$, $\alpha < \beta$ if and only if the $\alpha$th element comes before the $\beta$th element.