The Barber Paradox
The Barber paradox is attributed to the British philosopher Bertrand Russell. It highlights a fundamental problem in mathematics, exposing an inconsistency in the basic principles on which mathematics is founded.
The barber paradox asks us to consider the following situation:
In a village, the barber shaves everyone who does not shave himself, but no one else.
The question that prompts the paradox is this:
Who shaves the barber?
No matter how we try to answer this question, we get into trouble.
If we say that the barber shaves himself, then we get into trouble. The barber shaves only those who do not shave themselves, so if he shaves himself then he doesn’t shave himself, which is self-contradictory.
If we say that the barber does not shave himself, then problems also arise. The barber shaves everyone who does not shave himself, so if he doesn’t shave himself then he shaves himself, which is again absurd.
Even if we try to get clever, saying that the barber is a woman, we do not evade the paradox. If the barber is a woman, then she either shaves herself (and so is one of the people not shaved by the barber), or does not shave herself (and so is one of the people shaved by the barber).
Both cases, then, are impossible; the barber can neither shave himself nor not shave himself. The question ‘Who shaves the barber?’ is unanswerable.