The Debates on Infinity: A Mathematical History Approach

Abstract

Calculus and set theory sparked centuries of debate on infinity, which continues today. After discovering paradoxes and inconsistencies, mathematicians and philosophers questioned the underlying systems and conceptions of the infinite. Today, we easily use the infinity sign in our academic work. We often forget infinity's turbulent debut in our contemporary use of the term. David Hilbert wrote "On the Infinite" in the 1920s to persuade sceptics to embrace and use infinity. Gödel and his second incompleteness theorem defeated him very quickly. Beyond Gödel's claim of system inconsistency, Hilbert's theory neglected two factors when seen from a current viewpoint. First is the genuine nature of actual infinity, and second is Zermelo-Fraenkel's (ZF) axiom-based refined set theory. This article will look into the these great achievements and argue the deficiencies of them.