Russell’s paradox

E2517

antimony in naive set theory logical paradox set-theoretic paradox

Russell’s paradox is a foundational logical contradiction in naive set theory that reveals problems with sets that contain themselves, leading to major developments in modern logic and the axiomatization of set theory.

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Observed surface forms (2)

Surface form	Occurrences
Russell paradox	5
Russell's paradox	5

Statements (41)

Predicate	Object
instanceOf	antimony in naive set theory ⓘ logical paradox ⓘ set-theoretic paradox ⓘ
appearsIn	Principia Mathematica ⓘ surface form: "Principia Mathematica"
category	paradoxes in set theory ⓘ paradoxes of self-reference ⓘ
communicatedTo	Gottlob Frege ⓘ
concerns	naive set theory ⓘ self-membership of sets ⓘ the set of all sets that do not contain themselves ⓘ
discoveredBy	Bertrand Russell ⓘ
discoveryYear	1901 ⓘ
field	foundations of mathematics ⓘ mathematical logic ⓘ set theory ⓘ
formalizes	the contradiction arising from considering the set of all sets that are not members of themselves ⓘ
hasExampleFormulation	the set of all sets that are not members of themselves is a member of itself if and only if it is not a member of itself ⓘ
hasKeyQuestion	Does the set of all sets that are not members of themselves contain itself? ⓘ
impact	prompted rigorous foundations for mathematics ⓘ showed need to distinguish sets and proper classes ⓘ
influenced	Zermelo–Fraenkel set theory ⓘ the axiomatization of set theory ⓘ the development of modern logic ⓘ type theory ⓘ
ledTo	the development of Russell’s type theory ⓘ the introduction of restricted comprehension axioms ⓘ the separation axiom in Zermelo–Fraenkel set theory ⓘ
logicalForm	self-referential contradiction ⓘ
motivated	axiomatic set theory ⓘ
namedAfter	Bertrand Russell ⓘ
relatedTo	Barber paradox ⓘ Burali-Forti paradox ⓘ Cantor’s paradox ⓘ liar paradox ⓘ
resolutionApproach	restricting set formation by axioms ⓘ using type hierarchies to block self-membership ⓘ
shows	limitations of naive set theory ⓘ naive comprehension leads to contradiction ⓘ there is no set of all sets that are not members of themselves ⓘ unrestricted set formation is inconsistent ⓘ
undermined	Frege’s system in "Grundgesetze der Arithmetik" ⓘ

Referenced by (15)

Full triples — surface form annotated when it differs from this entity's canonical label.

Zermelo set theory → addressesProblem → Russell’s paradox ⓘ

this entity surface form: Russell paradox

Morse–Kelley set theory by class–set distinction → avoidsParadox → Russell’s paradox ⓘ

Barber paradox → basedOn → Russell’s paradox ⓘ

this entity surface form: Russell's paradox

Zermelo–Fraenkel set theory → designedToAvoid → Russell’s paradox ⓘ

this entity surface form: Russell paradox

Barber paradox → illustrates → Russell’s paradox ⓘ

this entity surface form: Russell's paradox

set theory → includesConcept → Russell’s paradox ⓘ

this entity surface form: Russell's paradox

Frege’s system in "Grundgesetze der Arithmetik" → inconsistencyRevealedBy → Russell’s paradox ⓘ

Curry paradox → isAnalogousTo → Russell’s paradox ⓘ

this entity surface form: Russell paradox

Bertrand Russell → knownFor → Russell’s paradox ⓘ

Barber paradox → relatedTo → Russell’s paradox ⓘ

this entity surface form: Russell's paradox

Berry paradox → relatedTo → Russell’s paradox ⓘ

Burali-Forti paradox → relatedTo → Russell’s paradox ⓘ

this entity surface form: Russell paradox

Cantor’s paradox → relatedTo → Russell’s paradox ⓘ

Epimenides paradox → relatedTo → Russell’s paradox ⓘ

this entity surface form: Russell paradox

liar paradox → relatedTo → Russell’s paradox ⓘ

this entity surface form: Russell's paradox