How did Russell’s theory of types restrict self-referential sets?

To address the logical contradictions found in the system of Gottlob Frege—specifically Russell’s paradox regarding the set of all sets that are not members of themselves—Bertrand Russell developed the theory of types.

This theory restricts self-referential sets by establishing a hierarchical structure for logical entities:

• Logical Types: Every set is assigned a specific logical type. A set of a certain type can only contain members of a lower type, which inherently prevents a set from containing itself as a member.1
• Vicious Circle Principle: Russell identified that paradoxes often arise from "impredicative" definitions, where an object is defined by a quantifier that includes the object itself. To prevent this, the theory prohibits the use of predicate functions with arguments of equal or higher type.21
• Ramified Hierarchy: In Principia Mathematica, Russell and Alfred North Whitehead further complicated this structure to avoid broader paradoxes, ensuring that definitions cannot rely on the totality to which the defined object belongs.1