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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