The axiom of choice and its applications: An exploration of the '100 prisoners problem'

Abstract

The Axiom of Choice (AC), a cardinal principle in set theory, postulates that for any assortment of disjoint non-empty sets, its possible to construct a new set by selecting one element from each set in the collection. Using choice functions, this idea suggests that every collection of nonempty sets can be associated with a choice function. In the mathematical landscape, ACs significance is accentuated by its extensive application in a myriad of mathematical deductions, marking it as a cornerstone among mathematical axioms. This study delves into the practical implications and applications of AC, employing rigorous analytical methods to investigate its vast and multifaceted influence on the broader mathematical domain. Key findings indicate that AC has facilitated the proof of various theorems, some of which, on the surface, appear unrelated. While its inception sparked considerable debates due to concerns over its intuitive validity, its impact on contemporary mathematics is profound. This research underscores ACs central role in advancing mathematical thought, highlighting its contributions to both foundational theories and intricate proofs.