Zorn's lemma ZornsLemma 2002-01-05 08:28:54 2009-02-16 15:42:14 Theorem Set theory %\usepackage{amssymb} %\usepackage{amsmath} %\usepackage{amsfonts} \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}.