Infinity Research Paper

Abstract: This paper is a report on the concept of infinity and its evolution throughout time. Although rather intriguing, difficult to conceptualize, or even clearly define, the idea of infinity has allowed for the progression of mathematics as a whole. What is infinity, what defines the different ideas of infinity, and who were the mathematicians who attempted to define it in an abstract, theoretical manner beneficial to mathematics?

Introduction

What exactly is infinity? Is it truly never ending, and in reality what does this actually mean? Since the beginning of time these have been questions of immeasurable debate throughout different cultures. The Greek, called it “aperion,” which meant “unbounded, infinite, indefinite, or undefined”

Pythagoras and his followers believed that everything in the “world could be expressed by an arrangement involving just the whole numbers” [3]. However, they were truly perplexed to discover that the diagonal of a square was incommensurable with its side. For example, a square with a side length of 1 would produce a diagonal with a length of √(2 ). This ratio could not be expressed by the whole numbers, and in fact, the ratio was a nonrepeating, nonterminating, decimal series [3]. In modern day mathematics, these numbers are known as irrational. Undoubtedly, this was one of the earliest appearances of infinity in

1. If, for example, we draw two concentric circles and proceed to connect, from the center of the innermost circle, lines to the outermost circle, we will discover a one-to-one correspondence with the points on the circles. This is paradoxical in the sense that we know the innermost circle contains and infinite amount of points, so does this imply that the outermost contains twice as many infinite points, in other words the first is infinity and the second, larger, twice infinity [3]? Then how does the one-to-one correspondence exist? Indeed, this is quite perplexing and Galileo Galilei came to the conclusion that when speaking of infinities, “we cannot speak of infinite quantities as being the one greater or less than or equal to another”