Paradoxes are fun. Metaphors are fun.

There is a nice metaphor for Russell’s paradox: suppose there is a town, and in this town, there is a barber who cuts the hair of every townsperson who doesn’t cut their own hair. Who cuts the barber’s hair?

This is a metaphor for a very technical mathematics phenomenon, so in a sense, laypeople not interested in mathematics can get a feel for what Russell’s paradox says.

An easy answer to the metaphor is that such a barber simply does not exist, but what does this mean for mathematics?

Well, let’s give some quick (naive) definitions of sets, functions, and predicates so we can derive this paradox and show that our set axioms produce inconsistencies.

A set is a well-defined collection of things. We can have a set of all people who drink alcohol, or a set of various boxes, or a set of numbers, or whatever.

So there is a thing called a universal set, which is defined to be the set of all sets. According to our definition of sets, this is a legit set. Of course, in our particular set theory, the universal set will lead to a paradox, but for now, we’ll remain naive and keep trucking along.

A function f:X \rightarrow Y for sets X and Y is a set of ordered pairs (x, y) such that (x, y_1), (x, y_2) \in f \Rightarrow (x, y_1) = (x, y_2) or the y value is uniquely determined up to x. Then we may write f(x) = y, which matches our intuition of how functions really work.

A predicate is a function p:X \rightarrow \{\mathrm{troof}, \mathrm{lolwut}\}. Suppose we have a set X with a predicate p on X. The set of all elements in X elements which evaluates to troof on the predicate p is denoted \{x \in X : p(x)\}.

Notice that according to our definitions so far, inclusion is actually a predicate on sets. That is, if we have a set X, then we have a predicate \in: X \rightarrow \{T, F\}, \in(x) = T if x is in X and \in(x) = F otherwise. The former case is notated x \in X, the latter x \notin X. Example: pizza \in \{foods\} but thermite \notin \{foods\}.

Let’s call our set of all sets V (our town). Then build a set W = \{x \in V : x \notin x\} (our barber, and the inclusion is analogically a haircut). Since V is the set of all sets, W \in V (our assumption that the barber is in the town). But is W an element of W? We have that W is an element of W if and only if W is not an element of W. That doesn’t make any sense!

So our naive definition of a set leads to a logical contradiction. For this particular paradox, we can probably blame self-reference as the culprit of the paradox.

Metaphorically, when we say that such a barber does not exist, we mean mathematically that such naive axiomatic systems do not exist because they lead to logical contradictions. Whether this is the truth is a philosophical discussion, though.