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, $\mathrm{\varnothing }$, 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

$0$$=$$\mathrm{\varnothing }$

$1$$=$$\left\{0\right\}=\left\{\mathrm{\varnothing }\right\}$

$2$$=$$\{0,1\}=\{\mathrm{\varnothing },\left\{\mathrm{\varnothing }\right\}\}$

$3$$=$$\{0,1,2\}=\{\mathrm{\varnothing },\left\{\mathrm{\varnothing }\right\},\left\{\{\mathrm{\varnothing },\left\{\mathrm{\varnothing }\right\}\}\right\}\}$

$\mathrm{⋮}$

$N$$=$$\{0,1,\mathrm{\dots },N-1\}$