The question of model companionability : positive and negative answers

Abstract

Model companion ofa universal theory T is the axiomatization of the existentially closed models of T. This thesis studies the concept of model companionability of theories. We present examples of model companions of certain well known theories. We then give examples of theories without model companions. The main focus of this thesis is to elaborate a technique, which we call "the Compactness Argument". Compactness Argument is used to prove that the model companion of a theory does not exist. We apply Compactness Argument to prove that the following theories do not have model companions: the theory of groups, the theory of rings, two examples of the theory of graphs, the theory of fields with two commuting automorphisms, and the theory of dense linear orders with an automorphism. Several proofs are illustrated by original diagrams to provide a better understanding to the reader.