natural number
Given the ZermeloFraenkel axioms^{} of set theory^{}, one can prove that there exists an inductive set^{} $X$ such that $\mathrm{\varnothing}\in X$. The natural numbers^{} $\mathbb{N}$ are then defined to be the intersection^{} of all subsets of $X$ which are inductive sets and contain the empty set^{} as an element.
The first few natural numbers are:

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$0:=\mathrm{\varnothing}$

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$1:={0}^{\prime}=\{0\}=\{\mathrm{\varnothing}\}$

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$2:={1}^{\prime}=\{0,1\}=\{\mathrm{\varnothing},\{\mathrm{\varnothing}\}\}$

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$3:={2}^{\prime}=\{0,1,2\}=\{\mathrm{\varnothing},\{\mathrm{\varnothing}\},\{\mathrm{\varnothing},\{\mathrm{\varnothing}\}\}\}$
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 $0,1,\mathrm{\dots},n1$, and $n$ is both a subset of $\mathbb{N}$ and an element of $\mathbb{N}$.
In some contexts (most notably, in number theory^{}), it is more convenient to exclude $0$ from the set of natural numbers, so that $\mathbb{N}=\{1,2,3,\mathrm{\dots}\}$. When it is not explicitly specified, one must determine from context whether $0$ is being considered a natural number or not.
Addition^{} of natural numbers is defined inductively as follows:

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$a+0:=a$ for all $a\in \mathbb{N}$

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$a+{b}^{\prime}:={(a+b)}^{\prime}$ for all $a,b\in \mathbb{N}$
Multiplication of natural numbers is defined inductively as follows:

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$a\cdot 0:=0$ for all $a\in \mathbb{N}$

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$a\cdot {b}^{\prime}:=(a\cdot b)+a$ for all $a,b\in \mathbb{N}$
The natural numbers form a monoid under either addition or multiplication. There is an ordering relation on the natural numbers, defined by: $a\le b$ if $a\subseteq b$.
Title  natural number 

Canonical name  NaturalNumber 
Date of creation  20130322 11:50:05 
Last modified on  20130322 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 
Synonym  $\mathbb{N}$ 
Related topic  InductiveSet 
Related topic  Successor^{} 
Related topic  PeanoArithmetic 
Related topic  VonNeumannInteger 