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

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.  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 }

4.  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

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

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

10. Thank you for attention