x INTRODUCTION the connections between these two approaches at the end of Chapter 25. Before turning to that work, we discuss an extremely general framework. The basic properties of abstract elementary classes are developed in Part II (chapters 5-8). In particular, we give Shelah’s presentation theorem which repre- sents every AEC as a pseudoelementary class (class of reducts to a vocabulary 𝐿 of a first order theory in an expanded language 𝐿′) that omit a set of types. Many of the key results (especially in Part IV) depend on having owenheim number ℵ0. Various successes and perils of translating 𝐿𝜔 1 ,𝜔 (𝑄) to an AEC (with count- able owenheim number) are detailed in Chapters 6-8 along with the translation of classes defined by sentences of 𝐿𝜔 1 ,𝜔 to the class of atomic models of a first or- der theory in an expanded vocabulary. Chapter 8 contains Shelah’s beautiful ZFC proof that a sentence of 𝐿𝜔 1 ,𝜔 (𝑄) that is ℵ1-categorical has a model of power ℵ2. In Part III (Chapters 9-17) we first study the conjecture that for ‘reasonably well-behaved classes’, categoricity should be either eventually true or eventually false. We formalize ‘reasonably well-behaved’ via two crucial hypotheses: amalga- mation and the existence of arbitrarily large models. Under these assumptions, the notion of Galois type over a model is well-behaved and we recover such fundamental notions as the identification of ‘saturated models’ with those which are ‘model ho- mogeneous’. Equally important, we are able to use the omitting types technology originally developed by Morley to find Ehrenfeucht-Mostowski models for AEC. This leads to the proof that categoricity implies stability in smaller cardinalities and eventually, via a more subtle use of Ehrenfeucht-Mostowski models, to a notion of superstability. The first goal of these chapters is to expound Shelah’s proof of a downward categoricity theorem for an AEC (satisfying the above hypothesis) and categorical in a successor cardinal. A key aspect of that argument is the proof that if 𝑲 is categorical above the Hanf number for AEC’s, then two distinct Galois types differ on a ‘small’ submodel. Grossberg and VanDieren [GV06c] christened this notion: tame. We refine the notion of tame in Chapter 11 and discuss three properties of Galois types: tameness, locality, and compactness. Careful discussion of these notions requires the introduction of cardinal parameters to calibrate the notion of ‘small’. We analyze this situation and sketch examples related to the White- head conjecture showing how non-tame classes can arise. Grossberg and VanDieren [GV06b, GV06a]develop the theory for AEC satisfying very strong tameness hy- potheses. Under these conditions they showed categoricity could be transferred upward from categoricity in two successive cardinals. Key to obtaining categoric- ity transfer from one cardinal 𝜆+ is the proof that the union of a ‘short’ chain of saturated models of cardinality 𝜆 is saturated. This is a kind of superstability consideration it requires a further and still more subtle use of the Ehrenfeucht- Mostowski technology and a more detailed analysis of splitting this is carried out in Chapter 15. In Chapters 16 and 17we conclude Part III and explore AEC without as- suming the amalgamation property. We show, under mild set-theoretic hypotheses (weak diamond), that an AEC which is categorical in 𝜅 and fails the amalgamation property for models of cardinality 𝜅 has many models of cardinality 𝜅+. In Part IV (Chapters 18-26) we return to the more concrete situation of atomic classes, which, of course, encompasses 𝐿𝜔 1 ,𝜔 . Using 2ℵ0 2ℵ1, one deduces from a theorem of Keisler [Kei71] that an ℵ1-categorical sentence 𝜓 in 𝐿𝜔 1 ,𝜔 is 𝜔-stable.
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