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

Bertrand Russel ( )  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

 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 }

 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

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

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

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

 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

 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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