Russell’s Paradox Made Easy

Russell’s Paradox Made Easy  [1]

Consider the set of all men: its members are men, though it is not itself a man but a set. Most sets are like this: they don’t have sets as members, but ordinary things—flowers, bees, motorcars. (Some sets do have sets as members, such as the set of sets with two members, but we can put these aside here.) A set is an abstract entity, a collection of concrete things in the case of men and bees. Thus sets are not usually members of themselves—they are collections of non-sets. There are exceptions, such as the set of all sets: that set is included in itself, because it is the set of all sets and it is itself a set. But this is not the typical case: nearly all sets don’t include themselves, since they are usually sets of objects that are not sets (such as men or bees). Call these sets that don’t include themselves “ordinary sets”: then we can say that the set of men is an ordinary set—the kind that doesn’t have itself as a member.  [2]

Now suppose that we consider all these ordinary sets: we collect them together into a single collection. This is a very big set, since there are vastly many ordinary sets of objects. Notice that it is a set that has sets as members, since it is the set of all ordinary sets—not the set of all the objects there are, such as men, bees, and motorcars. It is a bit like the set of all sets, except that it includes only ordinary sets, i.e. those that don’t have themselves as members. Anyway, we form this big set of ordinary sets, which doesn’t seem difficult—such a set surely exists. You can imagine a drawing of it as a circle containing lots of dots for all the members. Now we can ask a question: Is this set itself an ordinary set? Is it the kind of set that doesn’t include itself, like nearly all sets? Is the dot for it outside of the circle?

Suppose we say that it is an ordinary set; then it does not include itself among its own membership. It stands apart from its members. But then it must be included in itself, since it is the set of all ordinary sets. If it is an ordinary set, then it must belong in the set of all ordinary sets; but then it is not an ordinary set, because it includes itself in itself. It must be an exceptional set, like the set of absolutely every set: it must include itself among its members. Suppose instead that it does include itself. Then it is an exceptional set not an ordinary set. That means it is a member of itself, i.e. it is a set that falls within its own scope. But if it is a member of the set of all ordinary sets, then it must be ordinary; but it can’t be ordinary since it is a member of itself. Thus if the set of ordinary sets is a member of itself, it is not an ordinary set, while if it is not a member of itself it is an ordinary set. It has a choice: it can either be a member of itself or not, but if it is it is not and if it is not then it is. Thus the set of all ordinary sets is a contradictory set: it is neither one thing nor the other. If it’s ordinary it’s exceptional, but if it’s exceptional it’s ordinary. The problem is that the set that combines all ordinary sets faces a dilemma: if it’s ordinary it must include itself, in which case it is not ordinary; but if it doesn’t include itself, then it must be ordinary, in which case it must include itself.

            Ordinary sets like the set of men or the set of bees are not problematic at all: they are simply not members of themselves, not being men or bees but sets of men or bees. But if you collect all of these sets together to form one big set you face an awkward question, namely “What kind of set is that?” If you say it’s like the sets it has as members, then it will be among its members, but then it’s not an ordinary set; but if you say it’s not like these member sets, then it won’t be included alongside them, in which case it will be an ordinary set. If it’s ordinary, it includes itself, which makes it not ordinary; but if it’s not ordinary, then it includes itself, in which case it is a member of the set of ordinary sets. Either way you get a contradiction. And yet there is nothing amiss with ordinary sets as such—they are not contradictory—and there seems nothing objectionable about bringing them all together into one big set. So two harmless-looking things put together lead to a contradiction. That is Russell’s paradox. It is called a paradox because it goes from seemingly innocuous assumptions to an outright contradiction. Clearly there are sets of objects such as men and bees, and clearly these sets can join together to form a set consisting of all of them; but then we generate a contradiction from the nature of that set. What seemed self-evidently correct thus leads to logical inconsistency.

Colin McGinn     

  [1] I write this because I have never read an exposition of Russell’s paradox that is intuitive and accessible enough for a novice. They tend to be too concise and rigorous for an undergraduate or lay reader; I want to make the paradox as natural and comprehensible as possible.

  [2] The set of sets with two members is also an ordinary set, since it does not itself have two members, and hence is not a member of itself. Ordinary sets can be sets of individuals like the set of men or sets of sets like the set of sets with two members, but they don’t by definition contain themselves.