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Russel‘s paradox (also known as Russel‘s antimony)

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Bertrand Russel (1872-1970) British philosopher, logician, mathematician, historian and social critic Was awarded Nobel prize in literature Is known for challenging foundations of mathematics by discovering Russel’s paradox

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Shows that naive set theory leads to a contradiction According to this theory any definable collection is a set Let R be the set of all sets that are not members of themselves Symbolically: let R = {x | x ∉ x }

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If R is not a member of itself, then its definition dictates that it must contain itself If it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves Symbolically all together: let R = { x | x ∉ x }, then R ∈ R R ∉ R

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Applied versions From the list of versions that are closer to real-life situations, the barber paradox is the most famous

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Barber paradox Suppose there is a town with just one barber, who is a male

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In this town every man keeps himself clean-shaven, doing exactly one of these things: 1.shaving himself 2.going to the barber

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Asking the question, “who shaves the barber ?” results in paradox Both of the possibilities result in barber shaving himself, but this is not possible since he only shaves the men, who do not shave themselves

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There is no way to solve this paradox, it can only be avoided Most famous way to avoid this paradox is Zermelo-Fraenkel’s set theory In this set theory sets are constructed just using axioms

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Thank you for attention

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