BEYOND FIRST ORDER LOGIC: FROM NUMBER OF STRUCTURES TO STRUCTURE OF NUMBERS: PART II

We study the history and recent developments in nonelementary model theory focusing on the framework of abstract elementary classes. We discuss the role of syntax and semantics and the motivation to generalize first order model theory to nonelementary frameworks and illuminate the study with concrete examples of classes of models. This second part continues to study the question of catecoricity transfer and counting the number of structures of certain cardinality. We discuss more thoroughly the role of countable models, search for a non-elementary counterpart for the concept of completeness and present two examples: one example answers a question asked by David Kueker and the other investigates models of Peano Arithmetic and the relation of an elementary end-extension in terms of an abstract elementary class.