Russell’s paradox

Russell’s paradox, discovered by Bertrand Russell in 1901, was a paradox with set theory as it existed at the time. It struck at the heart of set theory and to some extent mathematics itself. This led to significant changes in set theory and the general formalisation of mathematics.

As an introduction, we will look at a simple paradox developed by Russell himself to help explain Russell’s paradox. We will then look at main problem and the most widely accepted resolution.

The barber’s paradox

The barber of a small village claims that he shaves every man in the village who doesn’t shave himself and that he doesn’t shave anyone else.

This might seem a reasonable claim as he is the only barber in the village.

But the question is, does he shave himself? That creates a paradox. If he doesn’t shave himself, that contradicts the statement that he shaves every man who doesn’t shave himself. If he does shave himself, then it contradicts the statement that he doesn’t shave any man who does shave himself.

In fact, not everyone regards this as a paradox. One interpretation is that the barber simply didn’t think to include himself as a special case. The barber either shaves himself or doesn’t, and he probably does, because who else would do it? Maybe the fact that he shaves himself was too obvious to mention.

But when we apply this to the mathematical properties of sets, it is a bit more difficult to wriggle out of.