# von Neumann integer

A von Neumann 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 (http://planetmath.org/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 integers (http://planetmath.org/Integer).

## Examples

 $\displaystyle 0$ $\displaystyle=$ $\displaystyle\emptyset$ $\displaystyle 1$ $\displaystyle=$ $\displaystyle\left\{0\right\}=\left\{\emptyset\right\}$ $\displaystyle 2$ $\displaystyle=$ $\displaystyle\left\{0,1\right\}=\left\{\emptyset,\left\{\emptyset\right\}\right\}$ $\displaystyle 3$ $\displaystyle=$ $\displaystyle\left\{0,1,2\right\}=\left\{\emptyset,\left\{\emptyset\right\},% \left\{\left\{\emptyset,\left\{\emptyset\right\}\right\}\right\}\right\}$ $\displaystyle\vdots$ $\displaystyle N$ $\displaystyle=$ $\displaystyle\left\{0,1,\dots,N-1\right\}$
Title von Neumann integer VonNeumannInteger 2013-03-22 12:32:34 2013-03-22 12:32:34 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 03E10 NaturalNumber VonNeumannOrdinal