Russel's paradox

The set of all sets that are not members of themselves ("the Russell set") is a necessary contradiction.

Formally,
Let the Russel set be defined as $R=\{x\mid x\notin x\}$.
then $R\in R\iff R\notin R$.

According to the unrestricted comprehension principle, for any sufficiently well-defined mathematical property, there is the set of all and only the objects that have that property.

If $R$ is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox.

Named after Bertrand Russel