There are many examples in mathematics of statements which, overtly or covertly, assert the existence of an element maximal in some ordered set (commonly, a family of sets under inclusion). The first section of this chapter addresses the question of the existence of maximal elements. This question cannot be answered without a discussion of Zorn's Lemma and the Axiom of Choice, and this necessitates an excursion into the foundations of set theory. It would be inappropriate to include here a full discussion of the role and status in mathematics of Zorn's Lemma and its equivalents. Rather we seek to complement the treatment in set theory texts of this important topic and, although our account is self-contained, it is principally directed at readers who have previously encountered the Axiom of Choice. En route, we provide belated justification for the arguments in 2.39, prove some intrinsically interesting results about ordered sets, and derive the results on prime and maximal ideals on which the representation theory in Chapter 11 rests. Those who do not wish to explore this foundational material but who do wish to study Chapter 11, may without detriment, skip over the first section of this chapter; see 10.15.

Do maximal elements exist? – Zorn's Lemma and the Axiom of Choice

Aside from the treatment of ordinals, ordered sets have traditionally played a peripheral role in introductory set theory courses.