# Zorn’s lemma

If $X$ is a partially ordered set^{}
such that every chain in $X$ has an upper bound,
then $X$ has a maximal element^{}.

Note that the empty chain in $X$ has an upper bound in $X$ if and only if $X$ is non-empty. Because this case is rather different from the case of non-empty chains, Zorn’s Lemma is often stated in the following form: If $X$ is a non-empty partially ordered set such that every non-empty chain in $X$ has an upper bound, then $X$ has a maximal element. (In other words: Any non-empty inductively ordered set has a maximal element.)

In ZF, Zorn’s Lemma is equivalent^{} to the Axiom of Choice^{} (http://planetmath.org/AxiomOfChoice).

Title | Zorn’s lemma |

Canonical name | ZornsLemma |

Date of creation | 2013-03-22 12:09:04 |

Last modified on | 2013-03-22 12:09:04 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 10 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 06A06 |

Classification | msc 03E25 |

Related topic | AxiomOfChoice |

Related topic | MaximalityPrinciple |

Related topic | HaudorffsMaximumPrinciple |

Related topic | ZornsLemmaAndTheWellOrderingTheoremEquivalenceOfHaudorffsMaximumPrinciple |

Related topic | EveryVectorSpaceHasABasis |

Related topic | TukeysLemma |

Related topic | ZermelosPostulate |

Related topic | KuratowskisLemma |

Related topic | EveryRingHasAMaximalIdeal |

Related topic | InductivelyOr |