viii INTRODUCTION

not the place for even a cursory survey of the development of stability theory over

the last 35 years. However, the powerful tools of the Shelah’s calculus of indepen-

dence and orthogonality are fundamental to the applications of model theory in the

1990’s to Diophantine geometry and number theory [Bou99].

Since the 1970’s Shelah has been developing the intersection of the second

and third streams described above: the model theory of the class of models of

a sentence in one of a number of ‘non-elementary’ logics. He builds on Keisler’s

work [Kei71] for the study of 𝐿𝜔1,𝜔 but to extend to other logics he needs a more

general framework and the Abstract Elementary Classes (AEC) we discuss below

provide one. In the last ten years, the need for such a study has become more

widely appreciated as a result of work on both such concrete problems as complex

exponentiation and Banach spaces and programmatic needs to understand ‘type-

definable’ groups and to understand an analogue to ‘stationary types’ in simple

theories.

Our goal here is to provide a systematic and intelligible account of some central

aspects of Shelah’s work and related developments. We study some very specific

logics (e.g. 𝐿𝜔1,𝜔) and the very general case of abstract elementary classes. The sur-

vey articles by Grossberg [Gro02] and myself [Bal04] provide further background

and motivation for the study of AEC that is less technical than the development

here.

An abstract elementary class (AEC) 𝑲 is a collection of models and a notion

of ‘strong submodel’ ≺ which satisfies general conditions similar to those satisfied

by the class of models of a first order theory with ≺ as elementary submodel.

In particular, the class is closed under unions of ≺-chains. A L¨owenheim-Skolem

number is assigned to each AEC: a cardinal 𝜅 such that each 𝑀 ∈ 𝑲 has a strong

submodel of cardinality 𝜅. Examples include the class of models of a ∀∃ first order

theory with ≺ as substructure, a complete sentence of 𝐿𝜔1,𝜔 with ≺ as elementary

submodel in an appropriate fragment of 𝐿𝜔1,𝜔 and the class of submodels of a

homogeneous model with ≺ as elementary submodel. The models of a sentence of

𝐿𝜔1,𝜔(𝑄) (𝑄 is the quantifier ‘there exists uncountably many’) fit into this context

modulo two important restrictions. An artificial notion of ‘strong submodel’ must

be introduced to guarantee the satisfaction of the axioms concerning unions of

chains. More important from a methodological viewpoint, without still further and

unsatisfactory contortions, the L¨ owenheim number of the class will be ℵ1.

In general the analysis is not nearly as advanced as in the first order case. We

have only approximations to Morley’s theorem and only a rudimentary development

of stability theory. (There have been significant advances under more specialized

assumptions such as homogeneity or excellence [GH89, HLS05] and other works

of e.g. Grossberg, Hyttinen, Lessmann, and Shelah.) The most dispositive result

is Shelah’s proof that assuming

2ℵ𝑛 2ℵ𝑛+1

for 𝑛 𝜔, if a sentence of 𝐿𝜔1,𝜔 is

categorical up to ℵ𝜔 then is categorical in all cardinals. Categoricity up to ℵ𝜔 is

essential [HS90, BK].

The situation for AEC is even less clear. One would like at least to show that

an AEC could not alternate indefinitely between categoricity and non-categoricity.

The strongest result we show here is implicit in [She99]. Theorem 15.13 asserts:

There is a Hanf number 𝜇 (not computed but depending only on the L¨owenheim

number of 𝑲) such that if an AEC 𝑲 satisfying the general conditions of Part 3

is categorial in a successor cardinal larger than 𝜇, it is categorical in all larger