Axiomatic Treatment of Rank in Infinite Sets

Abstract

1. Summary. In many branches of mathematics the notion of rank plays an important part. H. Whitney [3] made a detailed axiomatic investigation of rank and several related ideas. All sets considered by Whitney are finite. In the present note the axiomatic treatment of rank is extended to sets of any cardinal. In the special case of algebraic dependence of elements of a field with respect to a sub-field, similar questions have already been considered by Steinitz [2].