Document Type : Research Paper
University of Illinois at Chicago, USA
University of Helsinki, Finland
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.