Beyond first order logic: From number of structures to structure of numbers: Part II
John T. Baldwin, Tapani Hyttinen, Meeri Kesälä · 2013
Abstract. We study the history and recent developments in non-elementary classes. We discuss the role of syntax and semantics and the motivation to generalize first order model theory to non-elementary frameworks and illuminate the study with concrete ex-amples of classes of models. This second part continues to study the question of catecoricity transfer and counting the number of structures of certain cardi-nality. We discuss more thoroughly the role of countable models, search for a non-elementary counterpart for the concept of com-pleteness and present two examples: one example answers a ques-tion 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. Beyond First Order Logic: in number of structures to structure of numbers, Part I, we studied the basic concepts in non-elementary model theory, such as syntax and semantics, the languages Lλκ and the notion of a complete theory in first order logic (i.e., in the language Lωω), which determines an elementary class of structures. Classes of structures which cannot be axiomatized as the models of a first-order theory, but might have some other ‘logical ’ unifying attribute, are called non-elementary.