It follows directly from Shelah’s structure theory that if T is a classifiable theory, then the isomorphism type of any model of T is determined by the theory of that model in the language L∞,ω1. Leo Harrington asked if one could improve this to the logic L∞, In [S. Shelah, Characterizing an -saturated model of superstable NDOP theories by its L∞,-theory, Israel Journal of Mathematics 140 61–111] Shelah gives a partial positive answer, showing that for T a countable superstable NDOP (...) theory, two -saturated models of T are isomorphic if and only if they have the same L∞,-theory. We give here a negative answer to the general question by constructing two classifiable theories, each with 21 pairwise non-isomorphic models of cardinality 1, which are all L∞,-equivalent, a shallow depth 3 ω-stable theory and a shallow NOTOP depth 1 superstable theory. In the other direction, we show that in the case of an ω-stable depth 2 theory, the L∞,-theory is enough to describe the isomorphism type of all models. (shrink)

Let T be a complete, superstable theory with fewer than ${2^{\aleph_{0}}}$ countable models. Assuming that generic types of infinite, simple groups definable in T eq are sufficiently non-isolated we prove that ω ω is the strict upper bound for the Lascar rank of T.

Assume T is stable, small and Φ(x) is a formula of L(T). We study the impact on $T\lceil\Phi$ of naming finitely many elements of a model of T. We consider the cases of $T\lceil\Phi$ which is ω-stable or superstable of finite rank. In these cases we prove that if T has $ countable models and Q = Φ(M) is countable and atomic or saturated, then any good type in S(Q) is τ-stable. If $T\lceil\Phi$ is ω-stable and (bounded, 1-based or of (...) finite rank) with $I(T, \aleph_0) , then we prove that every good p ∈ S(Q) is τ-stable for any countable Q. The proofs of these results lead to several new properties of small stable theories, particularly of types of finite weight in such theories. (shrink)