Categoricity and Scott Families

Effective model theory is an area of logic that analyzes the effective content of the typical notions and results of model theory and universal algebra. Typical notions in model theory and universal algebra are languages and structures, theories and models, models and their submodels, automorphisms and isomorphisms, embeddings and elementary embeddings. In this paper languages, structures, and models are assumed to be countable. There are many ways to introduce considerations of effectiveness into the area of model theory or universal algebra. Here we will briefly explain considerations of effectiveness for theories and their models on the one hand, and for just structures on the other hand. Let us begin by considering effectiveness for theories and their models. From the model theoretic point of view, given a first order theory, one is interested in finding models for the theory with specific algebraic or modeltheoretic properties. In this sense theories are the basic objects in model theory. A natural way of introducing effectiveness is, therefore, to begin by considering decidable theories, i.e. ones whose theorems form a decidable (i.e. computable or recursive) set.