The theory of ordered abelian groups does not have the independence property

Abstract

We prove that no complete theory of ordered abelian groups has the independence property, thus answering a question by B. Poizat. The main tool is a result contained in the doctoral dissertation of Yuri Gurevich and also in P. H. Schmitt’s Elementary properties of ordered abelian groups, which basically transforms statements on ordered abelian groups into statements on coloured chains. We also prove that every n n -type in the theory of coloured chains has at most 2 n {2^n} coheirs, thereby strengthening a result by B. Poizat.