# well-ordering principle for natural numbers

Every nonempty set $S$ of natural numbers^{} contains a least element; that is, there is some number $a$ in $S$ such that $a\le b$ for all $b$ belonging to $S$.

Beware that there is another statement (which is equivalent^{} to the axiom of choice^{}) called the *well-ordering principle*. It asserts that every set can be well-ordered.

Note that the well-ordering principle for natural numbers is equivalent to the principle of mathematical induction (or, the principle of finite induction).

Title | well-ordering principle for natural numbers |
---|---|

Canonical name | WellorderingPrincipleForNaturalNumbers |

Date of creation | 2013-03-22 11:46:38 |

Last modified on | 2013-03-22 11:46:38 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 18 |

Author | CWoo (3771) |

Entry type | Axiom |

Classification | msc 06F25 |

Classification | msc 65A05 |

Classification | msc 11Y70 |

Related topic | MaximalityPrinciple |

Related topic | WellOrderedSet |

Related topic | ExistenceAndUniquenessOfTheGcdOfTwoIntegers |