viii INTRODUCTION not the place for even a cursory survey of the development of stability theory over the last 35 years. However, the powerful tools of the Shelah’s calculus of indepen- dence and orthogonality are fundamental to the applications of model theory in the 1990’s to Diophantine geometry and number theory [Bou99]. Since the 1970’s Shelah has been developing the intersection of the second and third streams described above: the model theory of the class of models of a sentence in one of a number of ‘non-elementary’ logics. He builds on Keisler’s work [Kei71] for the study of 𝐿𝜔 1 ,𝜔 but to extend to other logics he needs a more general framework and the Abstract Elementary Classes (AEC) we discuss below provide one. In the last ten years, the need for such a study has become more widely appreciated as a result of work on both such concrete problems as complex exponentiation and Banach spaces and programmatic needs to understand ‘type- definable’ groups and to understand an analogue to ‘stationary types’ in simple theories. Our goal here is to provide a systematic and intelligible account of some central aspects of Shelah’s work and related developments. We study some very specific logics (e.g. 𝐿𝜔 1 ,𝜔 ) and the very general case of abstract elementary classes. The sur- vey articles by Grossberg [Gro02] and myself [Bal04] provide further background and motivation for the study of AEC that is less technical than the development here. An abstract elementary class (AEC) 𝑲 is a collection of models and a notion of ‘strong submodel’ ≺ which satisfies general conditions similar to those satisfied by the class of models of a first order theory with ≺ as elementary submodel. In particular, the class is closed under unions of ≺-chains. A L¨owenheim-Skolem number is assigned to each AEC: a cardinal 𝜅 such that each 𝑀 ∈ 𝑲 has a strong submodel of cardinality 𝜅. Examples include the class of models of a ∀∃ first order theory with ≺ as substructure, a complete sentence of 𝐿𝜔 1 ,𝜔 with ≺ as elementary submodel in an appropriate fragment of 𝐿𝜔 1 ,𝜔 and the class of submodels of a homogeneous model with ≺ as elementary submodel. The models of a sentence of 𝐿𝜔 1 ,𝜔 (𝑄) (𝑄 is the quantifier ‘there exists uncountably many’) fit into this context modulo two important restrictions. An artificial notion of ‘strong submodel’ must be introduced to guarantee the satisfaction of the axioms concerning unions of chains. More important from a methodological viewpoint, without still further and unsatisfactory contortions, the L¨ owenheim number of the class will be ℵ1. In general the analysis is not nearly as advanced as in the first order case. We have only approximations to Morley’s theorem and only a rudimentary development of stability theory. (There have been significant advances under more specialized assumptions such as homogeneity or excellence [GH89, HLS05] and other works of e.g. Grossberg, Hyttinen, Lessmann, and Shelah.) The most dispositive result is Shelah’s proof that assuming 2ℵ𝑛 2ℵ𝑛+1 for 𝑛 𝜔, if a sentence of 𝐿𝜔 1 ,𝜔 is categorical up to ℵ𝜔 then is categorical in all cardinals. Categoricity up to ℵ𝜔 is essential [HS90, BK]. The situation for AEC is even less clear. One would like at least to show that an AEC could not alternate indefinitely between categoricity and non-categoricity. The strongest result we show here is implicit in [She99]. Theorem 15.13 asserts: There is a Hanf number 𝜇 (not computed but depending only on the L¨owenheim number of 𝑲) such that if an AEC 𝑲 satisfying the general conditions of Part 3 is categorial in a successor cardinal larger than 𝜇, it is categorical in all larger

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