Zorn's Lemma

Zorn's Lemma states that in certain structured collections (called partially ordered sets), if every chain (a sequence where each item is comparable to the next) has an upper boundary, then there is at least one maximal element—an item beyond which nothing else can be added without losing the structure. Essentially, it guarantees the existence of these "largest" or "most complete" elements under specific conditions. Zorn's Lemma is a fundamental tool in mathematics, especially for proving the existence of objects that are difficult to construct directly, such as bases in vector spaces or maximal ideals in algebra.