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
Canonical name	VonNeumannInteger
Date of creation	2013-03-22 12:32:34
Last modified on	2013-03-22 12:32:34
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	7
Author	mathcam (2727)
Entry type	Definition
Classification	msc 03E10
Related topic	NaturalNumber
Related topic	VonNeumannOrdinal