4 research outputs found
The Karp complexity of unstable classes
A class K of structures is controlled if, for all cardinals lambda, the
relation of L_{infty,lambda}-equivalence partitions K into a set of equivalence
classes (as opposed to a proper class). We prove that the class of doubly
transitive linear orders is controlled, while any pseudo-elementary class with
the omega-independence property is not controlled
Karp complexity and classes with the independence property
A class K of structures is controlled if for all cardinals lambda, the
relation of L_{infty,lambda}-equivalence partitions K into a set of equivalence
classes (as opposed to a proper class). We prove that no pseudo-elementary
class with the independence property is controlled. By contrast, there is a
pseudo-elementary class with the strict order property that is controlled
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
THE KARP COMPLEXITY OF UNSTABLE CLASSES
Abstract. A class K of structures is controlled if, for all cardinals λ, the relation of L∞,λ-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that the class of doubly transitive linear orders is controlled, while any pseudo-elementary class with the ω-independence property is not controlled. 560 revision:2000-10-31 modified:2000-10-31 1