Non-forking frames in abstract elementary classes

by Jarden and Shelah. [JrSh:875]
Annals Pure and Applied Logic, 2013
The stability theory of first order theories was initiated by Saharon Shelah in 1969. The classification of abstract elementary classes was initiated by Shelah, too. In several papers, he introduced non-forming relations. Later, Shelah (2009) [17, 11] introduced the good non-forking frame, an axiomatization of the non-forking notion. We improve results of Shelah on good non-forming grames, mainly by weakening the stability hypothesis in several important theorems, replacing it by the almost lambda-stability hypothesis: The number of types over a model of cardinality lambda is at most lambda^+ . We present conditions on K_lambda, that imply the existence of a model in K_{lambda^{+n}} for all n . We do this by providing sufficiently strong conditions on K_lambda, that they are inherited by a properly chosen subclass of K_{lambda^+} . What are these conditions? We assume that there is a `non-forking' relation which satisfies the properties of the non-forking relation on superstable first order theories. Note that here we deal with models of fixed cardinality lambda . While in Shelah (2009) [17,II] we assume stability in lambda, so we can use brimmed (=limit) models, here we assume almost stability only, but we add an assumption: The conjugation property. In the context of elementary classes, the superstability assumption gives the existence of types with well-defined dimension and the omega-stability assumption gives the existence and uniqueness of models prime over sets. In our context, the local character assumption is an analog to superstability and the density of the class of uniqueness triples with respect to the relation preccurlyeq is the analog to omega-stability.

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