Before we dive deeper into the fascinating world of ordinal numbers, we want to ask the following question:# Motivation: Is "Being a Set Element" ("$\in$") a Relation?
(related to Part: Ordinal Numbers)
As a reminder, for two sets $X$ and $Y$ a (binary) relation is a subset of the Cartesian product $X\times Y.$ By definition, the Cartesian product $X\times Y$ consists of ordered pairs $(a,b)$ with $a\in X$ and $b\in Y.$ Therefore, the ordered pair $(X,Y)$ cannot be an element of $X\times Y$, since this would mean that $X\in X$ and $Y\in Y$. We have seen that self-contained sets are forbidden in the Zermelo-Fraenkel set theory we have developed so far. Thus, unfortunately, the question has to be denied.
However, it would be great to be able to use the huge toolset we have developed for relations in order to study the properties of being contained "$\in$". This motivates the following definition.