Zorn’s lemma

If X is a partially ordered setMathworldPlanetmath such that every chain in X has an upper bound, then X has a maximal elementMathworldPlanetmath.

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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the Axiom of ChoiceMathworldPlanetmath (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