A natural number is any of the numbers 0, 1, 2, 3... that can be used to measure the size of finite sets. |

A natural number is any of the numbers 0, 1, 2, 3... that can be used to measure the size of finite sets. |

The Peano postulates essentially uniquely describe the set ℕ of natural numbers: |

The Peano postulates essentially uniquely describe the set of natural numbers, which is denoted by N (or more properly by the unicode character ℕ if your browser supports unicode display). |

One can inductively define an addition on the natural numbers by requiring a + (b + 1) = (a + b) + 1. This turns the natural numbers (ℕ,+) into a commutative monoid with neutral element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and can therefore be embedded in a group. The smallest group containing the natural numbers is the integers. |

One can inductively define an addition on the natural numbers by requiring a + (b + 1) = (a + b) + 1. This turns the natural numbers (N, +) into a commutative monoid with neutral element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and can therefore be embedded in a group. The smallest group containing the natural numbers is the integers. |

Analogously, a multiplication * can be defined via a * (b + 1) = ab + a. This turns (ℕ,*) into a commutative monoid; addition and multiplication are compatible which is expressed in the [distribution law]?: |

Analogously, a multiplication * can be defined via a * (b + 1) = ab + a. This turns (N, *) into a commutative monoid; addition and multiplication are compatible which is expressed in the [distribution law]?: |

Natural numbers can be used for two purposes: to describe the position of an element in an ordered sequence, which is generalized by the concept of ordinal number, and to specify the size of a finite set, which is generalized by the concept of cardinal number. In the finite world, these two concepts coincide: the finite ordinals are equal to ℕ as are the finite cardinals. When moving beyond the finite, however, the two concepts diverge. |

Natural numbers can be used for two purposes: to describe the position of an element in an ordered sequence, which is generalized by the concept of ordinal number, and to specify the size of a finite set, which is generalized by the concept of cardinal number. In the finite world, these two concepts coincide: the finite ordinals are equal to N as are the finite cardinals. When moving beyond the finite, however, the two concepts diverge. |