How did Russell’s theory of types try to prevent self-referential set paradoxes?

Bertrand Russell’s theory of types was designed to resolve logical paradoxes, such as Russell’s paradox, which arise from the unrestricted use of predicate functions and the assumption that one can form a set of all sets that do not contain themselves.123

To prevent these self-referential or "impredicative" definitions, Russell introduced a hierarchical system of types. This structure dictates that:

• Hierarchy of Elements: Entities are organized into a hierarchy where a set of a specific type can only contain members of a lower type.23
• Restricted Membership: The expression $x\in X$ is only meaningful if $X$ is of an appropriate higher type than $x$. This prevents a set from being a member of itself, as a set cannot be of a higher type than itself.23
• Vicious Circle Principle: Russell sought to avoid circularity by ensuring that definitions proceed from more basic elements to more inclusive sets, thereby excluding definitions that rely on the totality to which the defined object belongs.23

While this approach successfully avoided contradictions, it was often criticized for being cumbersome, leading to the development of alternative systems like Zermelo-Fraenkel set theory.23