Russell’s Paradox is a logical paradox about sets, discovered by the philosopher and mathematician Bertrand Russell in 1901.

## What is a Set?

A set is any collection of objects or items. The elements of a set can be physical objects, such as apples or teapots, or abstract concepts, such as numbers or mathematical functions. There can be a finite or infinite number of elements in a set, or no elements. A set can be defined either by listing all the elements, such as {1, 12, 43} or {red, white, blue, purple}, or by using a rule or description, such as “the set of prime numbers”, “the set of factors of 24”, or “the set of people who live in my house”.

There is nothing paradoxical about a set whose definition contradicts itself. The set of prime numbers divisible by 10 is a perfectly valid example of a set; it’s just empty.

Items of any kind can be members of sets, even sets themselves. The set of all sets is one example, and the set of all sets with exactly four elements is another.

## Russell’s Paradox

The paradox Russell discovered involves the set of all sets that do not contain themselves as a member. Let’s call this set R. Some sets are members of R, such as the set of all cats: this set is not itself a cat, so it is not a member of the set of all cats, so it does not contain itself. Whereas the set of everything that is not a cat is itself not a cat, so it is a member of the set of all non-cats, so it does contain itself, so it is not a member of R.

But does R contain itself or not?

If it does, then it does not fit the description of sets that do not contain themselves (because it does contain itself); but if it doesn’t fit the description, then it’s not a member of R, and therefore it *doesn’t* contain itself, which is a contradiction.

Similarly, if R does not contain itself, then it does fit the description of sets that do not contain themselves (because it doesn’t contain itself); but if it fits the description, then it is a member of R, and therefore it *does* contain itself, which is a contradiction again.

So if R contains itself, then it doesn’t; and if it doesn’t contain itself, then it does! It is paradoxical in the same way as “This sentence is not true”: if the sentence is true, then (as it truthfully states) it is *not* true, whereas if the sentence is false, it is falsely making the claim that it is not true, so it *is* true.

## The Barber Paradox

The barber paradox is an alternative formulation of Russell’s paradox which is less abstract and may be easier to understand. Imagine a town full of men who keep their faces clean-shaven. Some shave themselves, and others get a barber to shave them. Suppose there is a barber who shaves all the men who do not shave themselves (and only those men). Does he shave himself or not? If he doesn’t, then he counts as one of the men who do not shave themselves, so he qualifies to be one of his customers, so he does shave himself. If he does shave himself, then he doesn’t count as one of the men who do not shave themselves, so he doesn’t qualify to be one of his customers, so he can’t shave himself.

## Conclusion

Russell tried to solve the paradox by saying that R, the set of sets that do not contain themselves, cannot exist. (This is not the same as the set being empty. For example, the set of square circles is empty, but it is a valid set that exists.)

Russell developed a theory of “types”, which designate levels in a hierarchy of things that exist. Physical objects, numbers, and so on are on the first level. Sets containing things from the first level are on the second level, sets containing sets from the second level are on the third level, and so on. Russell stipulated that the definition of a set cannot mix levels. He wrote:

The principle which enables us to avoid illegitimate totalities may be stated as follows: ‘Whatever involves all of a collection must not be one of the collection.’

## Quiz