2 research outputs found
Categoricity of abstract classes with amalgamation
Let K be an abstract elementary class with amalgamation, and Lowenheim Skolem
number LS(K). We prove that for a suitable Hanf number chi_0 if chi_0 <
lambda_0 <= lambda_1, and K is categorical in lambda^+_1 then it is categorical
in lambda_0
Introduction to: classification theory for abstract elementary class
Classification theory of elementary classes deals with first order
(elementary) classes of structures (i.e. fixing a set T of first order
sentences, we investigate the class of models of T with the elementary submodel
notion). It tries to find dividing lines, prove their consequences, prove
"structure theorems, positive theorems" on those in the "low side" (in
particular stable and superstable theories), and prove "non-structure,
complexity theorems" on the "high side". It has started with categoricity and
number of non-isomorphic models. It is probably recognized as the central part
of model theory, however it will be even better to have such (non-trivial)
theory for non-elementary classes. Note also that many classes of structures
considered in algebra are not first order; some families of such classes are
close to first order (say have kind of compactness). But here we shall deal
with a classification theory for the more general case without assuming
knowledge of the first order case (and in most parts not assuming knowledge of
model theory at all).
The present paper includes an introduction to the forthcoming book on
Classification Theory for Abstract Elementary Classe