An essay on model theory (Q1407204)

The purpose of this paper is to introduce model theory and its key ideas to people unfamiliar with logic. Accordingly, the author, starting from the basic definitions of formula, theory, model and so on, rapidly reaches and discusses crucial notions, such as definable set, type, monster model and the various levels of stability. Some emphasis is also reserved to independence; forking, regularity and modularity are dealt with here. Interactions with diophantine geometry (like Hrushovski's proof of the Mordell-Lang Conjecture for function fields), or possible attempts of a model-theoretic treatment of analytic structures are also discussed. The final part of the paper describes some possible developments in model theory, mostly concerning the author's research, aiming at better understanding the internal structure of (monster) models, or some piece of them. In this perspective, some matters like profinite structures, or a more advanced topological analysis of the space of types, and their motivations, are illustrated.