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.