A characterization of inaccessible cardinals

Abstract

A cardinal number which is too large to be reached by some process is generally said to be inaccessible by that process. Many kinds of inaccessible cardinals have been discussed and for a general survey the book of H. Bachmann [1, Chapter 7] may be consulted. We consider here two inaccessibility properties. We shall denote the cardinal of a set X by |X|. The first inaccessibility property will be called regularity: the cardinal| X| will be said to be regular if there does not exist a disjoint cover {X1: i ε I} of X such that(i)|X1|<|X|, for each i in I, and(ii)|I|<|X|.