Russell's paradox

A contradiction in naive set theory: the set of all sets that are not members of themselves belongs to itself if and only if it does not.

Naive set theory: An informal approach to sets that permits broad, unrestricted set formation. Its unrestricted comprehension principle allows the set that generates the contradiction.
Georg Cantor: A German mathematician who founded modern set theory and developed transfinite arithmetic. His work on sets formed the setting in which foundational contradictions became pressing.
Russell's letter to Frege: A 1902 letter in which Bertrand Russell informed Gottlob Frege of the paradox. Frege acknowledged its damage to his system in an added appendix.
Russell's barber paradox: A popular analogy in which a barber shaves exactly those who do not shave themselves. It mirrors the paradox's self-referential membership condition in a familiar setting.
Axiom schema of unrestricted comprehension: A principle asserting that every property determines a set of all objects satisfying it. Applying it to non-self-membership produces Russell's contradictory set.
Gottlob Frege: A German logician whose formal system aimed to derive arithmetic from logic. Russell's discovery exposed a contradiction in Frege's foundational system.
Principia Mathematica: A three-volume work by Bertrand Russell and Alfred North Whitehead that sought to derive mathematics from logic. Its ramified type theory was designed to avoid paradoxes of self-reference.
Burali–Forti paradox: A contradiction arising from the supposed set of all ordinal numbers. It shows that unrestricted collections produce contradictions beyond self-membership.
Self-membership: The relation in which a set is one of its own members. The paradox turns on whether the constructed set has this relation to itself.
Bertrand Russell: A British philosopher and logician who discovered and published the set-theoretic paradox bearing his name. He communicated the contradiction to Frege in 1902.

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