How Do the Axioms of Choice and Determinacy Create Notable Consequences Across Multiple Areas of Mathematics and to What Extent Is the Relatively Dominant Role of Choice Justified by These Consequences?

This paper provides an evaluation of the incompatibility of the Axioms of Choice and Determinacy within the Zermelo-Fraenkel set theory. This paper explores this discrepancy via observation of Gale-Stewart games on the Baire Space. By analyzing the effects caused by this discrepancy on multiple major fields of mathematics, a clear contrast between the strictness of the results of Determinacy and the pathological anomalies as a byproduct of the more permissible nature of Choice is observed. Ultimately, it is argued that the dominance of Choice over Determinacy, as seen throughout the more mainstream fields of Set Theory and Linear Algebra, is merely the result of the need for contextual consideration of each axiom’s utility in each relevant field.