A Simple and Elementary Proof of Zorn's Lemma
arXiv:2305.10258 · doi:10.47443/dml.2023.143
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 machinery. However, the statement of Zorn's Lemma itself uses only elementary terminology for partially ordered sets. Therefore, it is worthy to establish a proof using only such elementary terminology. Following this line of study, we give a new simple proof of Zorn's Lemma, which does not even use the notion of a well-ordered set.
4 pages; v3 corresponds to the journal version; (v4) reference added (see the acknowledgement in the manuscript)