When Einstein Walked with Gödel: Excursions to the Edge of Thought

by Jim Holt

To make matters a little less abstract, you might pretend that we live in a world with an infinite number of people. Now consider all the possible clubs (sets of people) that might exist in this world. The least exclusive of these clubs—the universal club—will be the one of which absolutely everyone is a member. The most exclusive—the null club—will be the one that has no members at all. In between these two extremes will lie an infinity of other clubs, some with lots of members, some with few. How big is this infinity? Is there any way of matching up clubs and people one to one, thereby showing that the two infinite collections are the same size? Suppose that every person can be paired off with precisely one club, and vice versa. Some of the people will happen to be members of the club with which they are paired (for example, the person who gets paired with the universal club). Others will happen not to be members of their associated club (for example, the person who gets paired with the null club). Those people make up a group you might call the Groucho Club. The Groucho Club is a sort of salon des refusés: it consists of all the people who are paired with clubs that won’t have them as members. So the person who gets paired with the null club—which, of course, excludes him—at least has the consolation of being a member of the Groucho Club. Now, here is where things get interesting. Since the pairing of people with clubs is assumed to be complete, there must be some fello...