Annals of Pure and Applied Logic 164 (3):135-191 (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-forking relations. Later, Shelah [17, II] introduced the good non-forking frame, an axiomatization of the non-forking notion.We improve results of Shelah on good non-forking frames, mainly by weakening the stability hypothesis in several important theorems, replacing it by the almost λ-stability hypothesis: The number of types over a model of cardinality λ is at most λ+.We present conditions on Kλ, that imply the existence of a model in Kλ+n for all n. We do this by providing sufficiently strong conditions on Kλ, that they are inherited by a properly chosen subclass of Kλ+. 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 a fixed cardinality, λ.While in Shelah [17, II] we assume stability in λ, so we can use brimmed 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 ω-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 ≼bs is the analog to ω-stability
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DOI 10.1016/j.apal.2012.09.007
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References found in this work BETA

The Amalgamation Spectrum.John T. Baldwin, Alexei Kolesnikov & Saharon Shelah - 2009 - Journal of Symbolic Logic 74 (3):914-928.
Linear Orderings.Joseph G. Rosenstein - 1983 - Journal of Symbolic Logic 48 (4):1207-1209.
Classification Theory and the Number of Nonisomorphic Models.S. Shelah - 1982 - Journal of Symbolic Logic 47 (3):694-696.

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Citations of this work BETA

Forking and Superstability in Tame Aecs.Sebastien Vasey - 2016 - Journal of Symbolic Logic 81 (1):357-383.
Building Independence Relations in Abstract Elementary Classes.Sebastien Vasey - 2016 - Annals of Pure and Applied Logic 167 (11):1029-1092.
Canonical Forking in AECs.Will Boney, Rami Grossberg, Alexei Kolesnikov & Sebastien Vasey - 2016 - Annals of Pure and Applied Logic 167 (7):590-613.
Shelah's Eventual Categoricity Conjecture in Universal Classes: Part I.Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (9):1609-1642.
Downward Categoricity From a Successor Inside a Good Frame.Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (3):651-692.

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