Quantifying Over Countable Sets: Positive vs Stationary Logic

Publisher Summary The study of extensions of first order predicate calculus lacks a program or an ideology. Such an ideology is usually provided by—patterns of possible theorems; hard open problems and hard theorems; and applications to other fields of mathematics or other sciences. For first order predicate calculus, the first and second aspect overlap: completeness and compactness theorem, Lowenheim and Hanf number calculations, categoricity theorems and definability theory give a good frame, and applications in algebra and non-standard analysis are well known. For extensions of predicate calculus a first try is to mimic classical model theory. Fewer hard theorems exist and hard open problems are scattered without much coherence. This chapter reviews results on countably compact extensions of L ѡѡ which have grown out of joint work with S. Shelah and J. Stavi. The chapter surveys the model theory of L p ww and L ww( aa) ,as far as they are parallel and exhibits their known differences and presents a modest application to topology.