Russell's paradox

Russell's paradox is a paradox in naive set theory that is named after Bertrand Russell, who described it in 1901. Ernst Zermelo discovered it about a year earlier. The paradox highlights some problems that set theory had at the time. When Georg Cantor developed the theory in the 1880s, he already suspected that his theory would lead to such problems.

Consider the set of sets that do not contain themselves. The paradox arises when we consider whether this set should contain itself. If it should, then the set no longer satisfies its condition. If it should not, then it should contain itself. Thus, a contradiction arises. This paradox, stemming from a simple set definition, exposed a flaw in set theory and led mathematicians to introduce restrictions on set creation, as described in axiomatic set theory.

About 20 years later, Russell came up with the Barber paradox, which says: "Suppose there is a barber who shaves all adult men who do not shave themselves." In this case, answering the question of whether the barber shaves himself is impossible. If he does, then he would not, because he "shaves all men who do not shave themselves." Similarly, if he does not, then he would. This leads to a contradiction.

As a result, Russell extended set theory by introducing types.