Cardinal Arithmetic

Abstract

One result of our study of the comparative sizes of sets will be to define a new concept, called cardinal number, and to associate with each set X a cardinal number, denoted by card X. The definitions are such that for each cardinal number a there exist sets A with card A = a. We shall also define an ordering for cardinal numbers, denoted as usual by ≦. The connection between these new concepts and the ones already at our disposal is easy to describe: it will turn out that card X = card Y if and only if X ~ Y, and card X < card Y if and only if X ≺ Y. (If a and b are cardinal numbers, a < b means, of course, that a ≦ b but a ≠ b.)