
We now turn our attention to the issue of whether there is one mother of all universal sets. Before reading any further, consider this for a moment. That is, is there one largest set that all other sets are a subset of? Or, in other words, is there a set of all sets? To help wrap our heads around this issue, consider the following riddle, known as the Barber of Seville Paradox.

In Seville, there is a barber who shaves all those men, and only those men, who do not shave themselves. Who shaves the barber?

Problem 3.24 In the Barber of Seville Paradox, does the barber shave himself or not?

Problem 3.24 is an example of a paradox. A paradox is a statement that can be shown, using a given set of axioms and definitions, to be both true and false. Recall that an axiom is a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved. Paradoxes are often used to show the inconsistencies in a flawed axiomatic theory. The term paradox is also used informally to describe a surprising or counterintuitive result that follows from a given set of rules. Now, suppose that there is a set of all sets and call it $$\mathcal{U}$$. That is, $$\mathcal{U}\coloneqq \{A\mid A\mbox{ is a set}\}$$.

Given our definition of $$\mathcal{U}$$, explain why $$\mathcal{U}$$ is an element of itself.

If we continue with this line of reasoning, it must be the case that some sets are elements of themselves and some are not. Let $$X$$ be the set of all sets that are elements of themselves and let $$Y$$ be the set of all sets that are not elements of themselves.

[prob:russell] Does $$Y$$ belong to $$X$$ or $$Y$$? Explain why this is a paradox.

The above paradox is one way of phrasing a paradox referred to as Russell’s Paradox, named after British mathematician and philosopher Bertrand Russell (1872–1970). How did we get into this mess in the first place?! By assuming the existence of a set of all sets, we can produce all sorts of paradoxes. The only way to avoid these types of paradoxes is to conclude that there is no set of all sets. That is, the collection of all sets cannot be a set itself.