chapter  12
14 Pages

Transfinite Arithmetic

WithGarrett Birkhoff, Saunders Mac Lane

This chapter concerns the relationship between numbers and sets. This is the cardinal approach to the positive integers, as contrasted with the ordinal approach exemplified by the Peano postulates, which regard position in the familiar sequence "one, two, three, four,." as basic. Developed with care, this cardinal approach enables one to define numbers in terms of sets, thereby reducing the totality of undefined terms which must be assumed in mathematics. The chapter's objective is to extend the cardinal approach so as to give a precise definition of infinite cardinal numbers, which play a basic role in modern mathematics. Using this definition, the chapter shows how to add, multiply, and raise to powers arbitrary cardinal numbers, showing in the process that these operations have most (though not all) of the properties possessed by the corresponding operations on positive integers.