Zorn’s lemma
If is a partially ordered set such that every chain in has an upper bound, then has a maximal element.
Note that the empty chain in has an upper bound in if and only if 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 is a non-empty partially ordered set such that every non-empty chain in has an upper bound, then 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 |