natural number

Given the Zermelo-Fraenkel axiomsMathworldPlanetmath of set theoryMathworldPlanetmath, one can prove that there exists an inductive setMathworldPlanetmath X such that X. The natural numbersMathworldPlanetmath are then defined to be the intersectionMathworldPlanetmath of all subsets of X which are inductive sets and contain the empty setMathworldPlanetmath as an element.

The first few natural numbers are:

  • 0:=

  • 1:=0={0}={}

  • 2:=1={0,1}={,{}}

  • 3:=2={0,1,2}={,{},{,{}}}

Note that the set 0 has zero elements, the set 1 has one element, the set 2 has two elements, etc. Informally, the set n is the set consisting of the n elements 0,1,,n-1, and n is both a subset of and an element of .

In some contexts (most notably, in number theoryMathworldPlanetmathPlanetmath), it is more convenient to exclude 0 from the set of natural numbers, so that ={1,2,3,}. When it is not explicitly specified, one must determine from context whether 0 is being considered a natural number or not.

AdditionPlanetmathPlanetmath of natural numbers is defined inductively as follows:

  • a+0:=a for all a

  • a+b:=(a+b) for all a,b

Multiplication of natural numbers is defined inductively as follows:

  • a0:=0 for all a

  • ab:=(ab)+a for all a,b

The natural numbers form a monoid under either addition or multiplication. There is an ordering relation on the natural numbers, defined by: ab if ab.

Title natural number
Canonical name NaturalNumber
Date of creation 2013-03-22 11:50:05
Last modified on 2013-03-22 11:50:05
Owner djao (24)
Last modified by djao (24)
Numerical id 16
Author djao (24)
Entry type Definition
Classification msc 03E10
Classification msc 74D99
Related topic InductiveSet
Related topic SuccessorMathworldPlanetmathPlanetmathPlanetmath
Related topic PeanoArithmetic
Related topic VonNeumannInteger