# 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

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

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

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

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

$\mathrm{\vdots}$ | ||||

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

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 |