# Membership Relation is Not Reflexive

## Theorem

Let $\Bbb S$ be a set of sets in the context of pure set theory

Let $\RR$ denote the membership relation on $\Bbb S$:

$\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \RR \iff a \in b$

$\RR$ is not in general a reflexive relation.

## Proof

In the extreme pathological edge case:

$S = \set S$

it is seen that:

$S \in S$

and so:

$\forall x \in S: x \in x$

demonstrating that $\RR$ is reflexive in this specific case.

However, in this case $\set S$ is a set on which the Axiom of Foundation does not apply.

This is seen in Set is Not Element of Itself.

Hence this set is not supported by Zermelo-Fraenkel set theory.

Consider the set:

$T = \set {\O, \set \O}$

Then we immediately see that:

$\O \notin \O$

and so $\RR$ is seen to be not reflexive.

$\blacksquare$