History of the axiom of choice

This paper examines the evolution of the Axiom of Choice (AC) in set theory and mathematics. The focus is on its independence from Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). The work of Georg Cantor, the originator of set theory, is examined. His well-ordering theorem was the basis for the proposal of the Axiom of Choice by Ernst Zermelo in 1904. Zermelo's statement of the AC stirred much debate in the mathematical community because of its implicit nature. Paradoxes resulted from the use of the AC which added more fuel to the debate and added pressure on Zermelo to present an explicit proof of the independence of the AC from ZFC. Zermelo was unsuccessful in proving its independence from ZFC. Kurt Godel was successful in establishing the consistency of the AC using his Second Incompleteness Theorem. Paul J. Cohen, using his method of forcing, finally formulated the proof of the independence of the AC from ZFC. This discovery catapulted set theory from a foundation of mathematics to a legitimate branch of modern mathematics with the AC as a fundamental aspect. This paper examines the exciting journey of the Axiom of Choice looking at the work of some of the most influential figures in contemporary mathematics.