Definition:Epsilon Relation

Definition
In the language of set theory $\in$, the membership primitive, is neither a class nor a set, but a primitive predicate.

To simplify formulations, it is useful to introduce a class which behaves identically to the standard membership relation $\in$ for sets.

This class, denoted $\Epsilon$, will be referred to as the epsilon relation.

In class-builder notation:


 * $\Epsilon := \set {\tuple {x, y}: x \in y}$

Thus, explicitly, $\Epsilon$ is a relation, taking arguments from ordered pairs of sets $x$ and $y$.

It consists of precisely those ordered pairs $\paren {x, y}$ satisfying $x \in y$.

The behaviour is thus seen to be identical to regular membership with sets.

It is not the same as class membership, because $x$ and $y$ must be set variables.

Also see

 * Epsilon Relation is Strictly Well-Founded
 * Definition:Standard Structure