Categoricity in restricted classes

Abstract

It is well known that only a few structures can be described up to isomorphism by their elementary theories in the class of all models of fixed cardinality. Unfortunately, most of the natural and interesting structures fail to have this property. The situation changes if we restrict the class of models by adding some more requirements. S.Tennenbaum [4] was the first to prove the standard model of arithmetics is finitely axiomatizable and categorical in the class of all computable models. Later on, A.Morozov [1, 3], A.Nies [2], and others proved categoricity and finite axiomatizability of some other structures in natural restricted classes. In my talk I will present a survey of results on categoricity and finite axiomatizability of some structures in restricted classes of models and outline main ideas of proofs. It is remarkable that these results essentially use computability theory. © Springer-Verlag Berlin Heidelberg 2005.