An Explanation to the Concept of Actual Infinity and Potential Infinity through Set Theory and Calculus

Abstract

The concept of infinity refers to either an unending process or a limitless quantity. Aristotle introduced two types of infinity: potential infinity and actual infinity. Potential infinity refers to a never-ending process, and actual infinity refers to a collection containing infinitely many elements. This paper presents a descriptive study of the concept of infinity and discusses its properties through set theory and calculus. Infinity plays a central role in the formation and development of mathematics, specifically in limit, derivative, and integral. Moreover, the similarities and differences between potential infinity and actual infinity are explained with the help of set theory and integral differential calculus. The relationship between mathematics and infinity is a vital one. Since infinity is an elusive and contradictory idea without mathematical tools, it is hard to understand it, and there is no other knowledge to explain and make it understandable. By the way, in the absence of infinity, mathematics will never survive. This paper provided some examples to show that without employing mathematics, solving problems involving infinity based on human intuitions or weak induction may provide inaccurate results or lead to contradictions. Therefore, this paper suggested that using mathematical tools is essential for solving problems involving infinity.