Zorn’s Lemma

Suppose a partially ordered set has the property that every chain (i.e. totally ordered subset) has an upper bound. Then the set contains at least one maximal element.

Zorn’s lemma is equivalent to the well-ordering theorem and the axiom of choice, in the sense that any one of them, together with the Zermelo–Fraenkel axioms of set theory, is sufficient to prove the others.

Zorn’s lemma is equivalent (in ZF) to three main results:

    Hausdorff maximal principle
    Axiom of choice
    Well-ordering theorem

Moreover, Zorn’s lemma (or one of its equivalent forms) implies some major results in other mathematical areas. For example,

    Banach’s extension theorem which is used to prove one of the most fundamental results in functional analysis, the Hahn–Banach theorem
    Every vector space has a Hamel basis, a result from linear algebra
    Every commutative unital ring has a maximal ideal, a result from ring theory
    Tychonoff’s theorem in topology

In this sense, we see how Zorn’s lemma can be seen as a powerful tool, especially in the sense of unified mathematics.