How does the barber paradox illustrate Russell’s paradox?

The barber paradox serves as a popular, intuitive illustration of Russell’s paradox, which exposed a fundamental inconsistency in early set theory.12

In the barber paradox, a barber claims to shave all those—and only those—who do not shave themselves. This creates an impossible situation: if the barber shaves himself, he violates his rule of only shaving those who do not; if he does not shave himself, he must shave himself according to his own rule.2

This mirrors the logical structure of Russell’s paradox, which concerns the set of all sets that do not contain themselves as members ($R=\{x|x\notin x\}$). The paradox arises when asking if this set $R$ is a member of itself:
* If $R$ is a member of itself, it must satisfy the condition of not being a member of itself.
* If $R$ is not a member of itself, it satisfies the condition of being a member of itself.1

Both examples demonstrate the dangers of unrestricted self-reference and circular definitions in formal logic.12