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): $R = \{\forall s \in R. ¬P(s) \iff s \not\in s\}$