Zorn's Lemma - Equivalent Forms of Zorn's Lemma

Equivalent Forms of Zorn's Lemma

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

1. Hausdorff maximal principle
2. Axiom of choice
3. Well-ordering theorem.

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

1. Banach's extension theorem which is used to prove one of the most fundamental results in functional analysis, the Hahn–Banach theorem
2. Every vector space has a Hamel basis, a result from linear algebra (to which it is equivalent)
3. Every commutative unital ring has a maximal ideal, a result from ring theory
4. Tychonoff's theorem in topology (to which it is also equivalent)

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