Generalized archimedean groups

Abstract

Introduction. In a previous paper [3], the concept of archimedean was generalized so as to obtain a larger class of groups that were referred to as regularly ordered. As it was shown in [3 ], these groups, unlike the archimedean ones, admit a formalization in the lower predicate calculus (LPC) of formal logic; yet they cannot be distinguished from archimedean groups by any properties formalizable in the LPC. Thus they can serve as an LPC-substitute for archimedean groups. This result was, however, based on several theorems stated without proof. It is our aim now to supply the proofs of these theorems. Another objective of this paper consists in giving a unified algebraic approach to regularly groups, as a counterpart to the metamathematical study presented in [3]. Throughout this paper, the term ordered group stands for totally (and, hence, torsion-free) additive abelian other than {O }. Such a group, A, is said to be discretely or densely according as it does, or does not, contain a smallest positive element (called its unit, i). A subset SCA is called an interval if the relations a 0 (briefly, n-regular), if every infinite interval of A contains at least one (and, hence, infinitely many) elements divisible by n. (1) If this holds for every n, we simply say that A is regularly ordered. A is referred to as regularly discrete (regularly dense) if its ordering is both regular and discrete (regular and dense, respectively). From this definition we immediately derive the following corollary, to be used later.