x INTRODUCTION
the connections between these two approaches at the end of Chapter 25. Before
turning to that work, we discuss an extremely general framework.
The basic properties of abstract elementary classes are developed in Part II
(chapters 5-8). In particular, we give Shelah’s presentation theorem which repre-
sents every AEC as a pseudoelementary class (class of reducts to a vocabulary 𝐿
of a first order theory in an expanded language 𝐿′) that omit a set of types. Many
of the key results (especially in Part IV) depend on having owenheim number
ℵ0. Various successes and perils of translating 𝐿𝜔1,𝜔(𝑄) to an AEC (with count-
able owenheim number) are detailed in Chapters 6-8 along with the translation
of classes defined by sentences of 𝐿𝜔1,𝜔 to the class of atomic models of a first or-
der theory in an expanded vocabulary. Chapter 8 contains Shelah’s beautiful ZFC
proof that a sentence of 𝐿𝜔1,𝜔(𝑄) that is ℵ1-categorical has a model of power ℵ2.
In Part III (Chapters 9-17) we first study the conjecture that for ‘reasonably
well-behaved classes’, categoricity should be either eventually true or eventually
false. We formalize ‘reasonably well-behaved’ via two crucial hypotheses: amalga-
mation and the existence of arbitrarily large models. Under these assumptions, the
notion of Galois type over a model is well-behaved and we recover such fundamental
notions as the identification of ‘saturated models’ with those which are ‘model ho-
mogeneous’. Equally important, we are able to use the omitting types technology
originally developed by Morley to find Ehrenfeucht-Mostowski models for AEC.
This leads to the proof that categoricity implies stability in smaller cardinalities
and eventually, via a more subtle use of Ehrenfeucht-Mostowski models, to a notion
of superstability. The first goal of these chapters is to expound Shelah’s proof of a
downward categoricity theorem for an AEC (satisfying the above hypothesis) and
categorical in a successor cardinal. A key aspect of that argument is the proof
that if 𝑲 is categorical above the Hanf number for AEC’s, then two distinct Galois
types differ on a ‘small’ submodel. Grossberg and VanDieren [GV06c] christened
this notion: tame.
We refine the notion of tame in Chapter 11 and discuss three properties of
Galois types: tameness, locality, and compactness. Careful discussion of these
notions requires the introduction of cardinal parameters to calibrate the notion
of ‘small’. We analyze this situation and sketch examples related to the White-
head conjecture showing how non-tame classes can arise. Grossberg and VanDieren
[GV06b, GV06a]develop the theory for AEC satisfying very strong tameness hy-
potheses. Under these conditions they showed categoricity could be transferred
upward from categoricity in two successive cardinals. Key to obtaining categoric-
ity transfer from one cardinal
𝜆+
is the proof that the union of a ‘short’ chain
of saturated models of cardinality 𝜆 is saturated. This is a kind of superstability
consideration; it requires a further and still more subtle use of the Ehrenfeucht-
Mostowski technology and a more detailed analysis of splitting; this is carried out
in Chapter 15.
In Chapters 16 and 17we conclude Part III and explore AEC without as-
suming the amalgamation property. We show, under mild set-theoretic hypotheses
(weak diamond), that an AEC which is categorical in 𝜅 and fails the amalgamation
property for models of cardinality 𝜅 has many models of cardinality
𝜅+.
In Part IV (Chapters 18-26) we return to the more concrete situation of atomic
classes, which, of course, encompasses 𝐿𝜔1,𝜔. Using
2ℵ0 2ℵ1
, one deduces from a
theorem of Keisler [Kei71] that an ℵ1-categorical sentence 𝜓 in 𝐿𝜔1,𝜔 is 𝜔-stable.
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