In Part 1 of the book, we provide a concrete example of the applicability of
the abstract methods to a mathematical problem. That is, we describe sufficient
conditions for categoricity in all powers and elaborate a specific algebraic example.
In Parts 2 and 3 we study more general contexts where only partial categoricity
transfer methods are known. We close the book in Part 4 by describing Shelah’s
solution of the categoricity transfer problem for 𝐿𝜔1,𝜔 and placing Part 1 in that
context.
One reason to study AEC is that first order logic is inadequate to describe
certain basic mathematical structures. One example is complex exponentiation;
the ring of integers is interpretable in (ℂ, ⋅, +, exp) with domain the zeros of the
sine function and so the resulting theory is unstable. Zilber suggested investigating
this structure in 𝐿𝜔1,𝜔. A somewhat simpler object, the covers of the multiplica-
tive group of ℂ, is a concrete example of a quasiminimal excellent classes. This
is a specific example of the notion of excellence discussed in Part 4. A notion of
excellence presupposes a notion of dependence. In the case of quasiminimality, this
dependence notion determines a geometry. We prove that a quasiminimal excellent
class is categorical in all uncountable powers. We expound several papers of Zilber
and explain the fundamental notion of excellence in the ‘rank one’ case. We study
the ‘covers’ situation in detail. It provides a ‘real’ example of categorical struc-
tures which are not homogenous and thus shows why the notion of excellence must
be introduced. We briefly discuss (with references) at the end of Chapter 3 the
extensions of these methods to other mathematical situations (complex exponen-
tiation and semi-abelian varieties). We return to the more general consideration
of categoricity of 𝐿𝜔1,𝜔 in Part 4 and discuss the full notion of excellence. After
Proposition 25.20, we examine which parts of Zilber’s work are implied by Shelah
[She83a, She83b] and the sense in which Zilber extends Shelah.
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