# 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