06A06-ZornsLemma.tex

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\pmtitle{Zorn's lemma}
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\pmtype{Theorem}
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%\pmkeywords{Set theory}
\pmrelated{AxiomOfChoice}
\pmrelated{MaximalityPrinciple}
\pmrelated{HaudorffsMaximumPrinciple}
\pmrelated{ZornsLemmaAndTheWellOrderingTheoremEquivalenceOfHaudorffsMaximumPrinciple}
\pmrelated{EveryVectorSpaceHasABasis}
\pmrelated{TukeysLemma}
\pmrelated{ZermelosPostulate}
\pmrelated{KuratowskisLemma}
\pmrelated{EveryRingHasAMaximalIdeal}
\pmrelated{InductivelyOr}

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\begin{document}
\PMlinkescapeword{equivalent}

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 \PMlinkname{Axiom of Choice}{AxiomOfChoice}.

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