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Zorn's Lemma is a statement about partially ordered sets. It states: * Every partially ordered set in which every chain has an upper bound contains at least one maximal element. ** (A chain is a totally ordered subset) This doesn't really sound too controversial. Suppose every chain does have an upper bound. Either that upper bound is a maximal element, or there's something "above" it. Extend the chain, lather, rinse, repeat. Either your chain won't have an upper bound (which is impossible becuase we've assumed /every/ chain has an upper bound, or we must eventually get a maximum. Well, not so fast. Things get hairy when you have uncountably infinite sized sets, and so things can go wrong. In fact, Zorn's Lemma is equivalent (using the usual set-theory background) to the Axiom of Choice, and that's not so obvious either.