INTRODUCTION xi
Note however that ๐-stability is proved straightforwardly (Chapter 7) if one assumes
๐ 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 โต๐, we prove a sentence of ๐ฟ๐1,๐ that is categorical up to
โต๐ is excellent. In Chapters 24-25 we report Lessmannโs [Les03] account of prov-
ing Baldwin-Lachlan style characterizations of categoricity for ๐ฟ๐1,๐ and Shelahโs
analog of Morleyโs theorem for excellent atomic classes. We conclude Chapter 25,
by showing how to deduce the categoricity transfer theorem for arbitrary ๐ฟ๐1,๐-
sentences from a (stronger) result for complete sentences. Finally, in the last chap-
ter we explicate the Hart-Shelah example of an ๐ฟ๐1,๐-sentence that is categorical
up to โต๐ 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 โต0. Thus, in particular, we have proved few substantive general
results concerning ๐ฟ๐1,๐(๐) (the existence of a model in โต2 is a notable exception).
Shelah has substantial not yet published work attacking the categoricity transfer
problem in the context of โframesโ; this work does apply to ๐ฟ๐1,๐(๐) and does
not depend on Lยจ owenheim number โต0. 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 Markerโs 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 โต๐ 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 Morleyโs 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.
