Russell's paradox

Russell's paradox is a paradox in naive set theory that is named after Bertrand Russell, who described it in 1901. German mathematician Ernst Zermelo discovered it in 1898, but did not publish the idea. 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.

The paradox is described as follows: Consider a set. This set contains all sets which do not contain themselves. Because this set is also a set, it must be considered whether or not this set contains itself. If it does not, then it is eligible to be placed in itself. If it does, then it should not be included. Thus, a paradox is observed.

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 is a way of describing the same problem in a different way. The Barber paradox 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 paradox.