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 𝐿𝜔 1 ,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 𝐿𝜔 1 ,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 𝐿𝜔 1 ,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.
Previous Page Next Page