Zorn’s Lemma and Some Applications

Zorn’s lemma is a result in set theory that appears in proofs of some non-constructive existence theorems throughout mathematics. We will state Zorn’s lemma below and use it in later sections to prove some results in linear algebra, ring theory, and group theory. In an appendix, we will give an application to metric spaces. The statement of Zorn’s lemma is not intuitive, and some of the terminology in it may be unfamiliar, but after reading through the explanation of Zorn’s lemma and then the proofs that use it you should be more comfortable with how it can be applied.