Classifiable theories without finitary invariants

Abstract

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 ∞ , ω 1 ( d . q . ) . Leo Harrington asked if one could improve this to the logic L ∞ , ℵ ϵ ( d . q . ) In [S. Shelah, Characterizing an ℵ ϵ -saturated model of superstable NDOP theories by its L ∞ , ℵ ϵ -theory, Israel Journal of Mathematics 140 (2004) 61–111] Shelah gives a partial positive answer, showing that for T a countable superstable NDOP theory, two ℵ ϵ -saturated models of T are isomorphic if and only if they have the same L ∞ , ℵ ϵ ( d . q ) -theory. We give here a negative answer to the general question by constructing two classifiable theories, each with 2 ℵ 1 pairwise non-isomorphic models of cardinality ℵ 1 , which are all L ∞ , ℵ ϵ ( d . q . ) -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 ∞ , ℵ ϵ ( d . q ) -theory is enough to describe the isomorphism type of all models.