INTRODUCTION xi Note however that ๐œ”-stability is proved straightforwardly (Chapter 7) if one assumes ๐œ“ 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 โ„ต๐œ”, we prove a sentence of ๐ฟ๐œ” 1 ,๐œ” that is categorical up to โ„ต๐œ” is excellent. In Chapters 24-25 we report Lessmannโ€™s [Les03] account of prov- ing Baldwin-Lachlan style characterizations of categoricity for ๐ฟ๐œ” 1 ,๐œ” and Shelahโ€™s analog of Morleyโ€™s theorem for excellent atomic classes. We conclude Chapter 25, by showing how to deduce the categoricity transfer theorem for arbitrary ๐ฟ๐œ” 1 ,๐œ” - sentences from a (stronger) result for complete sentences. Finally, in the last chap- ter we explicate the Hart-Shelah example of an ๐ฟ๐œ” 1 ,๐œ” -sentence that is categorical up to โ„ต๐‘› 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 Lยจ owenheim number โ„ต0. Thus, in particular, we have proved few substantive general results concerning ๐ฟ๐œ” 1 ,๐œ” (๐‘„) (the existence of a model in โ„ต2 is a notable exception). Shelah has substantial not yet published work attacking the categoricity transfer problem in the context of โ€˜framesโ€™ this work does apply to ๐ฟ๐œ” 1 ,๐œ” (๐‘„) and does not depend on Lยจ owenheim number โ„ต0. 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 Markerโ€™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 โ„ต๐œ” 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 Morleyโ€™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.
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