# Russell's paradox

> **Russell's paradox**: Let $$R$$ be the set of all sets that are not members of themselves. If $$R$$ is not a member of itself, then by definition it must contain itself. But if it does contain itself, then it contradicts its own definition.

The set $$R$$ contains objects that do not contain themselves. Since these objects are bound to be sets (because only sets can contain stuff), we can say that the set $$R$$ contains sets that are not members of itself.

Normally, a set does not contain itselves (that would lead to infinite recursion), so almost all sets are classified as belonging to $$R$$.

Now, we just need to determine where does the $$R$$ itself belongs:

* if $$R$$ is not a member of $$R$$, then his definition dictates that it must contain itself (because it has the required property).
* if $$R$$ is a member of $$R$$, then it contradicts its own definition (because it doesn't have the required property).

This paradox, like many others, emerges in connection with self-reference; "whenever there's a self-reference, a paradox lurks near by".

This paradox have been a forking point that spawned many set theories with different strategies on how to avoid it. A consistent system must impose restrictions on the set inclusion rules. The restriction of regulations about what can constitute a set, made way for the _axiomatic set theory_ i.e. _class theory_

In a more formal presentation,\
Russell's paradox states that\
the predicate $$P$$, "_a set containing itself_",\
holds for a set, $$X$$, if it does contain itself:\
$$P(X) \iff X \in X$$

So, the set $$X$$ contains members that are sets containing themselves:\
$$X = \{\forall x \in X.P(x) \iff x \in x \}$$

The set $$R$$ is a set whose members are the (normal) sets that don't contain themselves (they don't satisfy the predicate):\
$$R = \{\forall s \in R. ¬P(s) \iff s \not\in s\}$$
