## Mathematics Senior Capstone Papers

#### Document Type

Article

#### Publication Date

Spring 5-12-2020

#### Abstract

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.

#### Recommended Citation

McCormick, Bryan, "The Axiom of Choice and Related Topics" (2020). *Mathematics Senior Capstone Papers*. 23.

https://digitalcommons.latech.edu/mathematics-senior-capstone-papers/23