Reverse mathematics, Archimedean classes, and Hahn's Theorem
Rodney G. Downey, Reed Solomon · Cambridge University Press eBooks · 2017
Abstract. Archimedean classes and convex subgroups play important roles in the study of ordered groups. In this paper, we show that ACA0 is equivalent to the existence of a set of representatives for the Archimedean classes of an ordered abelian group. Hahn’s Theorem is the strongest known tool for classifying orders on abelian groups. It states that every ordered abelian group can be embedded into products of the additive group of the reals. We show that Hahn’s Theorem is also equivalent to ACA0. §1. Introduction. The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of ordinary mathematics. In this article, we consider Hahn’s Theorem, one of the central results of ordered abelian group theory. This article is self-contained with respect to the material on ordered groups (see Section 2), but the reader who is