Beyond First Order Logic: From number of structures to structure of numbers: Part I

We study the history and recent developments in nonelementarymodel theory focusing on the framework of abstractelementary classes. We discuss the role of syntax and semanticsand the motivation to generalize first order model theory to nonelementaryframeworks and illuminate the study with concrete examplesof classes of models. This first part introduces the main conceps and philosophies anddiscusses two research questions, namely categoricity transfer andthe stability classification.