Cardinal and Ordinal Arithmetic

Abstract

Let (a _i )_i∈I be a family of cardinals indexed by a set I. The cardinal sum or simply the sum Σ_i∈I a _i of (a _i )_i∈I is defined to be the cardinal of the union ∪_i∈I(a_ix). If a and b are cardinals then the sum a+ b of a and b is defined to be the cardinal of (a × 0) ∪ (b × 1). Notice that in these definitions the sets forming the unions are equipotent to the cardinals being added but (thanks to their second factors) are pairwise disjoint.