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 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
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