In this paper I will be discussing the Axiom of Choice and its equivalent statements. The Axiom of Choice is an axiom of Zermelo-Fraenkel set theory that states that given a collection of non-empty sets, there exists a choice function which selects one element from each set to form a new set. The equivalents of the Axiom of Choice that I will be discussing include Zorn’s Lemma, which states that a partially ordered set with every chain being bounded above contains a maximal element, and the Well-Ordering Theorem, which states that every set has a well ordering. In addition to proving the equivalence of these statements, I will be discussing the mathematics required to prove them individually, as well as each of their consequences across the field of mathematics.
McCormick, Bryan, "The Axiom of Choice and Related Topics" (2020). Mathematics Senior Capstone Papers. 23.