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 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
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