Boxes inside boxes

A box holding only an empty box is not empty: it holds one thing, the empty box. In set notation, $\varnothing$ has no elements, but $\{\varnothing\}$ has exactly one. And $\{\varnothing, \{\varnothing\}\}$ has two.

Starting from nothing at all, we can build as many different sets as we like:

$$0 = \varnothing, \quad 1 = \{\varnothing\}, \quad 2 = \{\varnothing, \{\varnothing\}\}, \quad 3 = \{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\}, \;\dots$$

Each set contains all the earlier ones, and the set we call $n$ has exactly $n$ elements. In a common approach to the foundations of mathematics, introduced by John von Neumann, these sets are the natural numbers. Remarkably, the whole of school mathematics can be built from the empty set in this way.

Can every description make a set? For a long time mathematicians assumed so. But in 1901 Bertrand Russell considered the description "the set $R$ of all sets that are not elements of themselves". Is $R$ an element of itself? If it is, then by its own description it is not; if it is not, then by its description it is. Either way we reach a contradiction. So that description does not define a set.

This discovery — Russell's paradox — showed that we need precise rules for which sets exist. The rules most mathematicians use today are the axioms of ZFC (Zermelo–Fraenkel set theory with the axiom of Choice). They let us build sets like the ones in the experiment, while forbidding descriptions like Russell's.