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.

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