Russell's Paradox

RUSSELL’S PARADOX

When I meet people and say I’m a mathematician I often get slightly strange responses. The usual ones are “Ooh, I’m no good at maths” or, more recently, “I wish I understood more about maths”. It’s funny how some people immediately boast about how bad they are at maths, but other people immediately try to show off how knowledgeable they are. I once met a guy at a wedding who immediately said “Doesn’t Russell’s paradox show that maths is a failure?” This was a particularly curious approach because someone who knows enough about maths to know this thing called Russell’s paradox would usually know enough to understand why this doesn’t mean maths is a failure. But of course he was the kind of guy who wanted to belittle me, probably because he felt inadequate.

Bertrand Russell (1872–1970) was a philosopher and mathematician (among other things). His paradox dates from 1901 and is also related to questions about keeping levels separate. It can be stated in informal terms like this.

Imagine a male barber in a town. The barber shaves all the men in the town who do not shave themselves, and nobody else. Who shaves the barber?

Now, the barber is supposed to shave all the men who do not shave themselves. So if the barber does not shave himself, then the barber is supposed to shave himself, and if he doesn’t shave himself, then he does. This is very tangled up. Let’s consider any man A in the town:

• If person A shaves person A, then the barber doesn’t shave person A.

• If person A does not shave person A, the barber shaves person A.

This results in a problem if person A is the barber, which is allowed because A represented any person in the town. In this case the two statements become:

• If the barber shaves the barber, then the barber doesn’t shave the barber.

• If the barber doesn’t shave the barber, then the barber shaves the barber.

Each of these statements produces a contradiction. This is Russell’s paradox. Formally, the paradox is stated in terms of sets instead. It says: consider the set S of all sets that are not members of themselves. Is this set a member of itself? If it is, then it isn’t. And if it isn’t, then it is. It’s a paradox.

The problem in this situation is not the logic per se but the very statement we started with. With the barber, we simply conclude that no such barber can exist. We have to do the same for the sets as well: no such set S can exist, which is to say that this is not a valid way of defining a set.

Russell’s paradox did not break the entirety of mathematics, contrary to what that guy I met at the wedding tried to claim. Rather, it highlights an important nuance that needs to be taken into account when we’re defining mathematical sets, which is that some descriptions allow sets that will result in a contradiction, so we have to be careful to rule that possibility out. This led to the very careful axiomatisation of set theory so that a set is not just “a collection of things” but is “a collection of things that can be defined by a particular list of constructions, and no others”. The technical goal of the axioms is basically to avoid Russell’s paradox. The idea is to say that we have different “levels” of sets, a bit like how we have different “levels” of logic. Russell’s paradox is caused by statements involving sets that loop back on themselves. As long as we have different levels, we can declare that a set of all sets is on a different level, and this prevents us making statements that loop back on themselves.1

This paradox, like many studied by mathematicians, might seem rather technical and abstract. However, it has helped me understand some crucial questions about tolerance in society.