             Why is Zorn’s Lemma plausible? Here is an informal argument. Given a nonempty inductive partially ordered set \((P,\le)\), pick an arbitrary element \(p_{0}\) of \(P\). If \(p_{0}\) is maximal, stop there. Otherwise pick an element \(p_{1} \gt p_{0}\); if \(p_{1}\) is maximal, stop there. Otherwise pick an element \(p_{2} \gt p_{1}\), and repeat the process. If none of the elements \(p_{0} \lt p_{1} \lt p_{2} \lt \ldots \lt p_{n} \lt \ldots\) is maximal, the \(p_{i}\) form a chain which, since \(P\) is inductive, has an upper bound \(q_{0}\). If \(q_{0}\) is maximal, stop there. Otherwise the procedure can be repeated with \(q_{0} \lt q_{1}\), …, and then iterated. This process must eventually terminate, since otherwise the union of the chains so generated would constitute a proper class, making \(P\) itself a proper class contrary to assumption. The point at which the process terminates yields a maximal element of \(P\).