 Zorn’s Lemma is the best-known principle ensuring the existence of such maximal elements.  To state it, we need a few definitions. Given a partially ordered set \((P, \le)\), an upper bound for a subset \(X\) of \(P\) is an element \(a\in P\) for which \(x\le a\) for every \(x\in X\); a maximal element of \(P\) may then be defined as an element \(a\) for which the set of upper bounds of \(\{a\}\) coincides with \(\{a\}\), which essentially means that no element of \(P\) is strictly larger than \(a\). A chain in \((P,\le)\) is a subset \(C\) of \(P\) such that, for any \(x\), \(y\in P\), either \(x\le y\) or \(y\le x\).  \(P\) is said to be inductive if every chain in \(P\) has an upper bound. Now Zorn’s Lemma asserts: