A simple and elementary proof of Zorn’s Lemma

Abstract

Zorn’s Lemma is a well-known equivalent of the Axiom of Choice. It is usually regarded as a topic in axiomatic set theory, and its historically standard proof (from the Axiom of Choice) relies on transfinite recursion, a non-elementary set-theoretic concept. However, the statement of Zorn’s Lemma itself uses only elementary terminology of partially ordered sets. Hence, it is worthy to establish a proof using only such elementary terminology. Following this line of study, a new simple proof of Zorn’s Lemma is given that does not even use the notion of a well-ordered set.