von Neumann integer

A von Neumann is not an integer, but instead a construction of a natural numberMathworldPlanetmath using some basic set notation. The von Neumann integers are defined inductively. The von Neumann integer zero is defined to be the empty setMathworldPlanetmath, , 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 theoryMathworldPlanetmath (for instance, Zermelo-Fraenkel set theoryMathworldPlanetmath). 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).


0 =
1 = {0}={}
2 = {0,1}={,{}}
3 = {0,1,2}={,{},{{,{}}}}
N = {0,1,,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