[[Set theory MOC]]
# Russell's paradox

**Russell's paradox** states that the **Russellian** [[Class]] $R$ defined by
$$
\begin{align*}
(\forall \shood x)[x \in R \iff x \notin x]
\end{align*}
$$
is not a [[set]]. #m/thm/set
For if it were a set, either $R \in R$ or $R \notin R$ implies its opposite, which is absurd.
The main issue at hand is unrestricted comprehension,
and different approaches to [[axiomatic set theory]] must take care to resolve this paradox.

## Related paradoxes

- [[Russell's paradox for categories]]

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