# What numbers are made of

According to how numbers (and more generally, ordinals) are defined by Zermelo and von Neumann, they are literally made of nothing.

In developing axioms for set theory, Zermelo used the sets 0, {0}, {{0}}, {{{0}}, {{{{0}}}} and so on as natural numbers. Here, 0 is the empty set, while S is an operator that defines the number following the input as the set that contains all the sets represented by the input.

In this way, numbers are for all intents and purposes derived by nesting empty sets within other sets.

Zermelo’s method is simple and works, but was later extended and refined by a more complex and useful definition of number developed by Von Neumann.