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INTRODUCTION xi Note however that 𝜔-stability is proved straightforwardly (Chapter 7) if one assumes 𝜓 has arbitrarily large models. In Chapters 18-23, we introduce an independence notion and develop excellence for atomic classes. Assuming cardinal exponentia- tion is increasing below ℵ𝜔, we prove a sentence of 𝐿𝜔 1 ,𝜔 that is categorical up to ℵ𝜔 is excellent. In Chapters 24-25 we report Lessmann’s [Les03] account of prov- ing Baldwin-Lachlan style characterizations of categoricity for 𝐿𝜔 1 ,𝜔 and Shelah’s analog of Morley’s theorem for excellent atomic classes. We conclude Chapter 25, by showing how to deduce the categoricity transfer theorem for arbitrary 𝐿𝜔 1 ,𝜔 - sentences from a (stronger) result for complete sentences. Finally, in the last chap- ter we explicate the Hart-Shelah example of an 𝐿𝜔 1 ,𝜔 -sentence that is categorical up to ℵ𝑛 but not beyond and use it to illustrate the notion of tameness. The work here has used essentially in many cases that we deal with classes with L¨ owenheim number ℵ0. Thus, in particular, we have proved few substantive general results concerning 𝐿𝜔 1 ,𝜔 (𝑄) (the existence of a model in ℵ2 is a notable exception). Shelah has substantial not yet published work attacking the categoricity transfer problem in the context of ‘frames’ this work does apply to 𝐿𝜔 1 ,𝜔 (𝑄) and does not depend on L¨ owenheim number ℵ0. We do not address this work [She0x, She00d, She00c] nor related work which makes essential use of large cardinals ([MS90, KS96]. A solid graduate course in model theory is an essential prerequisite for this book. Nevertheless, the only quoted material is very elementary model theory (say a small part of Marker’s book [Mar02]), and two or three theorems from the Keisler book [Kei71] including the Lopez-Escobar theorem characterizing well-orderings. We include in Appendix A a full account of the Hanf number for omitting types. In Appendix B we give the Keisler technology for omitting types in uncountable models. The actual combinatorial principle that extends ZFC and is required for the results here is the Devlin-Shelah weak diamond. A proof of the weak diamond from weak GCH below ℵ𝜔 appears in Appendix C. In Appendix D we discuss a number of open problems. Other natural background reference books are [Mar02, Hod87, She78, CK73]. The foundation of all this work is Morley’s theorem [Mor65a] the basis for transferring this result to infinitary logic is [Kei71]. Most of the theory is due to Shelah. In addition to the fundamental papers of Shelah, this exposition de- pends heavily on various works by Grossberg, Lessmann, Makowski, VanDieren, and Zilber and on conversations with Adler, Coppola, Dolich, Drueck, Goodrick, Hart, Hyttinen, Kesala, Kirby, Kolesnikov, Kueker, Laskowski, Marker, Medvedev, Shelah, and Shkop as well as these authors. The book would never have happened if not for the enthusiasm and support of Rami Grossberg, Monica VanDieren and Andres Villaveces. They brought the subject alive for me and four conferences in Bogota and the 2006 AIM meeting on Abstract Elementary Classes were essential to my understanding of the area. Grossberg, in particular, was a unending aid in finding my way. I thank the logic seminar at the University of Barcelona and especially Enriques Casanovas for the opportunity to present Part IV in the Fall of 2006 and for their comments. I also must thank the University of Illinois at Chicago and the National Science Foundation for partial support during the preparation of this manuscript.