A von Neumann integer is not an integer, but instead a construction of a natural number using some basic set notation. The von Neumann integers are defined inductively. The von Neumann integer zero is defined to be the empty set, $\emptyset$ and there are no smaller von Neumann integers. The von Neumann integer $N$ is then the set of all von Neumann integers less than $N$ The set of von Neumann integers is the set of all finite von Neumann ordinals.

This form of construction from very basic notions of sets is applicable to various forms of set theory (for instance, Zermelo-Fraenkel set theory). While this construction suffices to define the set of natural numbers, a little more work must be done to define the set of all integers.

Examples

\begin{eqnarray*} 0 & = & \emptyset \\ 1 & = & \left\{ 0 \right\} = \left\{ \emptyset \right\} \\ 2 & = & \left\{ 0, 1 \right\} = \left\{ \emptyset, \left\{ \emptyset \right\} \right\} \\ 3 & = & \left\{ 0, 1, 2 \right\} = \left\{ \emptyset, \left\{ \emptyset \right\}, \left\{ \left\{ \emptyset, \left\{ \emptyset \right\} \right\} \right\}\right\} \\ & \vdots & \\ N & = & \left\{ 0, 1, \dots, N-1 \right\} \end{eqnarray*}