von Neumann integerVonNeumannInteger2002-03-10 15:11:302004-03-08 16:07:39Definition\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}A \emph{von Neumann \PMlinkescapetext{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 \PMlinkname{von Neumann ordinals}{VonNeumannOrdinal}.
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 \PMlinkname{integers}{Integer}.
\subsubsection*{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*}