Motivation: Is Being a Set Element $"\in"$ a Relation?
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.
| | | | created: 2019-02-03 14:07:49 | modified: 2019-02-03 14:19:51 | by: bookofproofs | references: , 
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Bibliography (further reading)
 Ebbinghaus, H.-D.: “Einführung in die Mengenlehre”, BI Wisschenschaftsverlag, 1994, 3
 Hoffmann, D.: “Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise”, Hoffmann, D., 2018