Infinity

Abstract

This chapter provides an overview of the concept of infinity. Infinity occurs in many shapes and forms in mathematics. The points at infinity in projective geometry are very different from the infinite and infinitesimal quantities that occur in nonstandard analysis, or the transfinite numbers in set theory, or the infinity involved in a limiting process. To classify this variety, it is helpful to distinguish between actual infinity and potential infinity. According to the idea of actual infinity, infinite and finite quantities are subsumed under the same theory; an actually infinite quantity is just like a finite quantity, only bigger. According to the idea of potential infinity, infinity is merely a figure of speech. This chapter discusses the nature of infinity as it occurs in applied mathematics and physics, in both countable form (an infinite number of particles) and uncountable form (the continua of space and time and other real-valued physical quantities such as mass and charge). The concepts of actual infinity, absolute infinity, and potential infinity are explained in detail in the chapter.