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Russel‘s paradox

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

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  1. Russel‘s paradox (alsoknownasRussel‘santimony)

  2. 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

  3. Showsthatnaive set theoryleads to a contradiction • According to thistheoryanydefinablecollectionis a set • Let R bethe set ofallsetsthat are notmembersofthemselves • Symbolically: let R = {x | x ∉ x }

  4. If R isnot a memberofitself, thenitsdefinitiondictatesthatitmustcontainitself • Ifitcontainsitself, thenitcontradictsitsowndefinitionasthe set ofallsetsthat are notmembersofthemselves • Symbolicallyalltogether: let R = { x | x ∉ x }, then R ∈ R  R ∉ R

  5. Appliedversions • Fromthe list ofversionsthat are closer to real-lifesituations, thebarber paradox isthe most famous

  6. Barber paradox • Supposethereis a townwith just onebarber, whois a male

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

  8. 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

  9. Thereis 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 usingaxioms

  10. Thankyouforattention

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