Stability theory and Algebra

There are two origins for first-order theories. One type of theory arises by generalizing the common features of a number of different structures, e.g. the theory of groups, and formulating a set of axioms to encode these common features. Here the set of axioms is well understood, frequently it is finite or at least recursive, but there usually is no clear understanding of all the logical consequences of these axioms. The second type of theory arises by considering the set, T = Th(A), of all sentences true in a fixed structure A,3 e.g. the theory of arithmetic (N, +, 0) or the theory of the field of complex numbers (alias: the theory of algebraically closed fields of characteristic zero). The second case gives little more insight as to the truth in A (i.e. membership in T) of a given sentence ∅. But it does guarantee that for a given sentence ∅, either ∅ or ¬∅ is in T, that is, that T is a complete theory. When does a theory T of the first type, i.e. with well-understood axioms, posses this completeness property? An obvious sufficient condition is that T be secretly of the second type, that it have only one model, or, in jargon, T is categorical. Unfortunately (or fortunately depending on your point of view) for any theory with an infinite model, the Löwenheim-Skolem theorem shows this to be impossible: The theory has a model in every infinite power. In the mid-50's Łoś and Vaught discovered that if a theory T with no finite models is categorical in some infinite power α (all models with cardinality α are isomorphic) then T is complete. We will be dealing below with countable complete theories and will assume, unless stated to the contrary, that each theory has no finite models.