Shepherdson's Theorems for Fragments of Open Induction

Abstract

By a well-known result of Shepherdson, models of the theory IOpen (a first order arithmetic containing the scheme of induction for all quantier free formulas) are exactly all the discretely ordered semirings that are integer parts of their real closures. In this paper we prove several analogous results that provide algebraic equivalents to various fragments of IOpen.

Keywords

• weak arithmetics
• open induction
• linear induction
• real closed fields