Suppose there is a barber who shaves every man who doesn’t shave himself, and no man who does. Now ask: does the barber shave himself? Well, if he does, then he is someone he doesn’t shave, because we’ve said that he only shaves men who don’t shave themselves. Yet, if he doesn’t shave himself, then he is one of the men who he does shave, because we’ve said he shaves every man who doesn’t shave himself! Paradoxically then, he does and doesn’t shave himself!
This may sound like just a piece of silliness, but the philosopher Bertrand Russell (1872–1970) used it to show that paradoxes of this form have serious consequences for set theory (as well as for logic and mathematics). Set theory concerns classes (collections) and their members. Prior to Russell noticing this paradox, it had been considered true that there is a set (collection) of things corresponding to every description. However, here we seem to have a description ‘Barber who shaves all and only those who don’t shave themselves’ to which no set of things could corresponds. | |||