Classification Theory for Abstract Elementary Classes by Saharon Shelah, Paperback | Indigo Chapters

An abstract elementary class (AEC) is a class of structures of a fixed vocabulary satisfying some natural closure properties. These classes encompass the normal classes defined in model theory and naturalexamples arise from mathematical practice, e. g. in algebranot to mention first order and infinitary logics. An AEC is alwaysendowed with a special substructure relation which is not always the obvious one. Abstractelementary classes provide one way out of the cul de sac of the model theory of infinitarylanguages which arose from over-concentration on syntactic criteria. This is the second volume of a two-volume monograph on abstract elementary classes. It is quiteself-contained and deals with three separate issues. The first is the topic of universal classes, i. e. classes of structures of a fixed vocabulary such that a structure belongs to the class if andonly if every finitely generated substructure belongs. Then we derivefrom an assumption on the number of models, the existence of an (almost) good frame. The notion of frame is a natural generalization of the first order concept of superstability to this context. The assumptionsays that the weak GCH holds fora cardinal $\\lambda$, its successor and double successor, and the class is categorical in thefirst two, and has an intermediate value for the number of models in the third. In particular, we can conclude from this argumentthe existence of a model in the next cardinal. Lastly we deal with the non-structure part of thetopic, that is, getting many non-isomorphic models in the double successor of $ \\lambda$ underrelevant assumptions, we also deal with almost good frames themselves and somerelevant set theory. | Classification Theory for Abstract Elementary Classes by Saharon Shelah, Paperback | Indigo Chapters