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.
如果一个偏序集P中的每一个链C（即全序子集）都有上界x（x当然必须属于P），那么该偏序集P至少包含一个极大元。
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.
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