The barber paradox is a puzzle derived from Russell’s paradox, one of the most famous paradoxes in the foundation of set theory. It was suggested to the philosopher and mathematician Bertrand Russell as an illustration of his own paradox, but he deemed it to be little more than “noise without meaning”: a semantic paradoxClass of paradox involving not just mathematical or logical terms, but also semantic notions such as truth. rather than a logical paradox.[1][2]
The puzzle sets the scenario of a barber in a certain village who shaves all those, and only those, who do not shave themselves. The question posed is “Who shaves the barber?” If the barber shaves himself, then he is in violation of the rule that he only shaves those who do not shave themselves. And if he does not shave himself, then he falls under the category of those who do not shave themselves, meaning that he should shave himself. Thus such a barber cannot logically exist.[3]
The paradox arises because in set theory terms some classes have themselves as members, such as the class of all abstract objects, whereas others do not; the class of donkeys, for instance, is not itself a donkey. So there is a logical paradox when considering the class of all classes that are not members of themselves. If the barber shaves himself he is not in the class of those he only shaves, but if he does, then he is.[2]
Stated formally in terms of set theory, 
from which it can be shown that 
is a member of if, and only if, is not a member of .[4]
