RussellParadoxon

RussellParadoxon, commonly known as Russell's paradox, is a fundamental problem in naive set theory. It considers the set R of all sets that do not contain themselves. The question then is: does R contain itself? If it does, then by the defining property it must not be a member. If it does not, then it must contain itself. This leads to a contradiction, showing that unrestricted set formation is inconsistent in naïve theories.

Origin and formulation: The paradox was discovered by Bertrand Russell in 1901 and published in the Proceedings of the London Mathematical Society. It targeted Frege’s program to ground mathematics on logic using an unrestricted principle of comprehension, exposing a flaw in naive set-building procedures.

Impact and responses: Russell's paradox prompted a major reform of the foundations of mathematics. In axiomatic set theory, especially Zermelo–Fraenkel set theory (ZF/ZFC), the unrestricted principle is replaced by axioms that restrict set formation. The Axiom of Separation allows forming subsets only from already existing sets, preventing the existence of a universal set and self-reference. Type theory provides an alternative by organizing objects into a hierarchy of types that blocks self-containing definitions.

Legacy: The paradox remains a canonical example illustrating the need for rigorous foundations in mathematics and logic. It influenced the development of formal set theory, the study of logic, and philosophical discussions about the nature of sets, definitions, and truth. See also: Cantor’s paradox, Burali-Forti paradox, Frege's Grundgesetze.