The Mathematics of the Infinite

Abstract

THE MOST FAMILIAR EXAMPLE OF AN INFINITE COLLECTION in mathematics is the sequence of positive integers 1, 2, 3,⋯. There are many others, for example, the collection of all rational numbers, the collection of all circles in the plane, the collection of all spheres in space, etc. The idea of the infinite is implicit in many mathematical concepts, but it was not until the nineteenth century that the mathematical infinite became the subject of precise analysis. The first steps in this direction were taken by Bernard Bolzano (1781–1848) in the first half of that century, but his work went largely unnoticed at that time. The modern theory of the mathematical infinite—set theory—was created by Cantor in the latter half of the century. Although set theory initially encountered certain obstacles—which we shall discuss later—it has come to penetrate, and influence decisively, virtually every area of mathematics. It also plays a central role in the logical and philosophical foundations of mathematics.