Invariants for \(\omega\)-categorical, \(\omega\)-stable theories (Q1073013)

In this paper we give a complete solution to the classification problem for \(\omega\)-categorical, \(\omega\)-stable theories. More explicitly, suppose T is \(\omega\)-categorical, \(\omega\)-stable with fewer than the maximum number of models in some uncountable power. We associate with each model M of T a ''simple'' invariant \({\mathcal I}(M)\), not unlike a vector of dimensions, such that \({\mathcal I}(M)={\mathcal I}(N)\) if and only if \(M\cong N\). The spectrum function, I(-,T), for a first-order theory T is such that for all infinite cardinals \(\lambda\), I(\(\lambda\),T) is the number of nonisomorphic models of T of cardinality \(\lambda\). As an application of our ''structure theorem'' we determine the possible spectrum functions for \(\omega\)-categorical, \(\omega\)-stable theories.