How does Russell’s paradox arise from the set of all sets that are not members of themselves?

Russell’s paradox arises from the application of an unrestricted comprehension principle, which allows for the formation of a set based on any definable condition. Specifically, it involves the set $R$ of all sets that are not members of themselves, defined by the condition $x\notin x$.12

The paradox emerges when one asks whether $R$ is a member of itself:

• If $R$ is a member of itself ($R\in R$), it must satisfy the condition of its own definition, meaning it is not a member of itself ($R\notin R$).12
• If $R$ is not a member of itself ($R\notin R$), it satisfies the condition for membership in $R$, meaning it must be a member of itself ($R\in R$).12

This creates an impossible, self-contradictory situation. Bertrand Russell identified this as a vicious circle, demonstrating that an unfettered comprehension principle is logically inconsistent. Modern set theory resolves this by restricting how sets are constructed, ensuring that no set can be as large as the universal collection of all sets.13