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

Source 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.