Baseballs and Underpants

I promised to explain Russell’s set of all sets that are not members of themselves. This is great for me because I really have to struggle to make it clear, and I really enjoy a challenge.

Let’s limit ourselves to one room. In our kid’s bedroom we find four baseballs. A mathematician would call this collection of baseballs a set. This bunch of baseballs is not itself a baseball, so this set of baseballs is not a member of the set of baseballs.

Now we turn to underpants, stacked neatly in the dresser drawer. Is this stack, or set or underpants itself a pair of underpants? Of course not.

We do the same for socks, shirts, toy cars, pencils and paper clips until we empty the room.

Being of an artistic nature, we make an installation in the local art museum, and in the space the museum gives us, we stack the baseballs, underpants, socks, shirts, toy cars, pencils and paper clips. This fascinating artistic installation is a collection of sets—a set of sets. None of these sets is a member of itself—the group of toy cars is not a toy car—so the installation is the set of all sets (from our kid’s bedroom) that are not members of themselves.

But wait a minute. This installation, this set of sets—is it a member of itself? Well, it is a set, a collection of piles of underpants and paper clips, piles that are not themselves underpants or paperclips.

So what do we decide? If this is a set that deserves to be included in a description of the installation, then that would mean it was a member of the installation—a member of itself. But if that is the case, it can’t be in our installation, because the installation includes only sets that do not contain themselves.

On the other hand, if we banish the set of sets, then it is a set that does not contain itself, and so it is included—only to be excluded.

This argument dashed the hopes of mathematicians who set out to find and prove all theorems. And don’t feel too bad if you had trouble following the argument above. The implications drove some mathematicians crazy.

Every language, including mathematics and logic, has a sentence that says, “This sentence is false.”

Russell found a way out of his paradox by separating logical statements into types. 

A self-referent declaration is a different kind of assertion from true theorems and from false statements. A simple logic program would not know what to do with such a statement, but would most likely get tangled in an endless loop. We ourselves see that endless loop starting up, and that is the difference. We are capable of transcending endless loops, puzzling over them, and laughing them off. The computer has to run around in circles for a while until its software says, “That’s enough of that. Stop now.”

What does all this have to do with you and me? We self-aware creatures are unique the same way. Our self-reference just does not fit in a physical world of simple logic. When we are tootling along and not reflecting on the quality of our awareness, we fit in the physical universe. We are doing the same mental work as a computer would.

But our self-awareness, our ability to remember ourselves—when we exercise it—lifts us right out of the world.

Descartes argued that there were two types of entity—mind and body. This idea is called the Cartesian duality, and contemporary philosophers love to refute it with all sorts of models of consciousness based on structure. And they are right, and they completely miss the point.

It is true that there is no real split between mind and body. In theory a computer can do everything a mind can do, the vast majority of the time. But my self-awareness, my experience of my awake state, when I exercise it, is absolutely different from the sensible world. (You would say the same thing—I hope!—about yours.)

The only way the reductionist philosophers can be right—and they may well be—is way, way spookier than the duality they are trying to escape.

I love this stuff.