In many branches of mathematics, especially set theory, algebra, and topology, the Axiom of Choice (AC) is crucial and helps to enable the existence of choice functions for any arbitrary collection of non-empty sets without explicit construction. Examining the arguments both for and against the Axiom of Choice, this work addresses the paradoxes and challenges it presents, including the Banach-Tarski dilemma and problems in measure theory, so furthering mathematical theory. Although AC is commended for its theoretical contributions—especially in terms of enabling work with abstract and infinite sets—it is attacked by constructivist mathematicians who stress the need of specific approaches of proof. The study comes to the conclusion that, despite its non-constructive character and the disputes it causes, the Axiom of Choice stays a vital instrument in modern mathematics, so extending the limits of theoretical investigation and application. Its application should, however, be carefully considered in order to balance the needs for mathematical rigour and practical relevance with the advantages of abstraction.
