“0 is particularly nice. It's the class of sets equivalent to the set of all objects unequal to themselves! No object is unequal to itself, so 0 is the class of all empty sets. But all empty sets have the same members—none! So they’re not merely equivalent to each other—they’re all the same set. There's only one empty set! (A set is characterized by its membership list. There‘s no way to tell one empty membership list from another. Therefore all empty sets are the same thing!) Once I have the empty set, I can use a trick of von Neumann as an alter- native way to construct the number 1. Consider the class of all empty sets. This class has exactly one member: the unique empty set. It’s a singleton. “Out of nothing” I have made a singleton set—a “canonical representative” for the cardinal number 1. l is the class of all singletons—all sets with but a single element. To avoid circularity: “1 is the class of all sets equivalent to the set [ I I I.” In words, 1 is the class of all sets equivalent to the set whose only element is the empty set. Continuing, you get pairs, triplets, and so on. Von Neumann recursively constructs the whole set of natural numbers out of sets of sets of sets of nothing.”

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