INTRODUCTION ix

cardinals. This state of affairs in a major reason that this monograph is titled

categoricity. Although a general stability theory for abstract elementary classes is

the ultimate goal, the results here depend heavily on assuming categoricity in at

least one cardinal.

There are several crucial aspects of first order model theory. By Lindstr¨om’s

theorem [Lin69] we know they can be summarized as: first order logic is the only

logic (of Lindstr¨ om type) with L¨ owenheim number ℵ0 that satisfies the compactness

theorem. One corollary of compactness in the first order case plays a distinctive

role here, the amalgamation property: two elementary extensions of a fixed model

𝑀 have a common extension over 𝑀. In particular, the first order amalgamation

property allows the identification (in a suitable monster model) of a syntactic type

(the description of a point by the formulas it satisfies) with an orbit under the

automorphism group (we say Galois type).

Some of the results here and many associated results were originally developed

using considerable extensions to ZFC. However, later developments and the focus

on AEC rather than 𝐿𝜅,𝜔 (for specific large cardinals 𝜅 ) have reduced such reliance.

With one exception, the results in this book are proved in ZFC or in ZFC + 2ℵ𝑛

2ℵ𝑛+1 for finite 𝑛; we call this proposition the very weak generalized continuum

hypothesis VWGCH . The exception is Chapter 17which relies on the hypothesis

that

2𝜇

2𝜇+

for any cardinal 𝜇; we call this hypothesis the weak generalized

continuum hypothesis, WGCH. Without this assumption, some crucial results have

not been proved in ZFC; the remarkable fact is that such a benign assumption as

VWGCH is all that is required. Some of the uses of stronger set theory to analyze

categoricity of 𝐿𝜔1,𝜔-sentences can be avoided by the assumption that the class of

models considered contains arbitrarily large models.

We now survey the material with an attempt to convey the spirit and not the

letter of various important concepts; precise versions are in the text. With a few

exceptions that are mentioned at the time all the work expounded here was first

discovered by Shelah in a long series of papers.

Part I (Chapters 2-4) contains a discussion of Zilber’s quasiminimal excellent

classes [Zil05]. This is a natural generalization of the study of first order strongly

minimal theories to the logic 𝐿𝜔1,𝜔 (and some fragments of 𝐿𝜔1,𝜔(𝑄). It clearly

exposes the connections between categoricity and homogeneous combinatorial ge-

ometries; there are natural algebraic applications to the study of various expansions

of the complex numbers. We expound a very concrete notion of ‘excellence’ for a

combinatorial geometry. Excellence describes the closure of an independent 𝑛-cube

of models. This is a fundamental structural property of countable structures in a

class 𝑲 which implies that 𝑲 has arbitrarily large models (and more). Zilber’s

contribution is to understand the connections of these ideas to concrete mathe-

matics, to recognize the relevance of infinitary logic to these concrete problems,

and to prove that his particular examples are excellent. These applications require

both great insight in finding the appropriate formal context and substantial math-

ematical work in verifying the conditions laid down. Moreover, his work has led

to fascinating speculations in complex analysis and number theory. As pure model

theory of 𝐿𝜔1,𝜔, these results and concepts were all established in greater generality

by Shelah [She83a] more than twenty years earlier. But Zilber’s work extends She-

lah’s analysis in one direction by applying to some extensions of 𝐿𝜔1,𝜔. We explore