The Axiom of Choice, now used throughout non-foundational mathematics with minimal hesitation, has an extremely interesting history and many non-trivial consequences.

First formulated explicitly by Ernst Zermelo to solve the so-called Well-Ordering Problem, this axiom sparked debates throughout the mathematical community. David Hilbert, arguably one of the greatest mathematicians of his time, described Choice as the axiom "most attacked up to the present in the mathematical literature;" and in Gregory Moore's words "if the Axiom's most severe constructivist critics prevailed, mathematics would be reduced to a collection of algorithms."

Modern mathematics, however, severely relies on the Axiom, even if not in its full strength. Commonly used mathematical principles such as Zorn's Lemma, the trichotomy of cardinals, the right-invertibility of surjective functions, and the existence of a basis for arbitrary vector spaces are all consequences of, and, in fact, equivalent to, the Axiom of Choice.

Through this inaugural talk of the Student Logic and History of Math Seminar, we explore some of the history, philosophy, and mathematics surrounding the Axiom of Choice.