Russell’s paradox

Suppose that for any coherent propositionPlanetmathPlanetmathPlanetmath P(x), we can construct a set {x:P(x)}. Let S={x:xx}. Suppose SS; then, by definition, SS. Likewise, if SS, then by definition SS. Therefore, we have a contradictionMathworldPlanetmathPlanetmath. Bertrand Russell gave this paradoxMathworldPlanetmath as an example of how a purely intuitive set theoryMathworldPlanetmath can be inconsistent. The regularity axiom, one of the Zermelo-Fraenkel axiomsMathworldPlanetmath, was devised to avoid this paradox by prohibiting self-swallowing sets.

An interpretationMathworldPlanetmathPlanetmath of Russell paradox without any formal languageMathworldPlanetmath of set theory could be stated like “If the barber shaves all those who do not themselves shave, does he shave himself?”. If you answer himself that is false since he only shaves all those who do not themselves shave. If you answer someone else that is also false because he shaves all those who do not themselves shave and in this case he is part of that set since he does not shave himself. Therefore we have a contradiction.

Remark. Russell’s paradox is the result of an axiom (due to Frege) in set theory, now obsolete, known as the axiom of (unrestricted) comprehension, which states: if ϕ is a predicateMathworldPlanetmath in the languagePlanetmathPlanetmath of set theory, then there is a set that contains exactly those elements x such that ϕ(x). In other words, {xϕ(x)} is a set. So if we take ϕ(x) to be xx, we arrive at Russell’s paradox.

Title Russell’s paradox
Canonical name RussellsParadox
Date of creation 2013-03-22 11:47:49
Last modified on 2013-03-22 11:47:49
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 03-00
Synonym axiom of unrestricted comprehension
Related topic ZermeloFraenkelAxioms
Related topic LambdaCalculus
Defines axiom of comprehension