Beyond first order logic: From number of structures to structure of numbers: Part II

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 second part continues to study the question of catecoricitytransfer 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 completenessand present two examples: one example answers a questionasked by David Kueker and the other investigates models ofPeano Arithmetic and the relation of an elementary end-extensionin terms of an abstract elementary class.