An Investigation of the Fields of Bounded Formal Power Series by Means of Theory of Cuts

Abstract

We consider a class K of real closed fields F, |F|=|G|=ℵ1, where G is a group of Archimedean classes of F, and cofinality of each symmetric gap of F is ℵ1. We will show that this class is exactly a class of all bounded formal power series R〚G,ℵ1〛, where G is a divisible Abelian group, card(G)=ℵ1, under CH. A nonstandard real line *R, which is η1-set belongs to this class; we will also consider a construction R〚G(L,P),ℵ1〛 of fields from this class, where L is a totally ordered set, P is a totally ordered field, G(L,P) is a group of finite words. It will be describes symmetric gaps of such two fields in K, which are not η1-set.