Classifiable theories without finitary invariants

: 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,1 (d.q.). Leo Harrington asked if one could improve this to the logic L, (d.q.) In [Sh 04] S. 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, (d.q)-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, (d.q.)equivalent: a shallow depth 3 -stable theory and a shallow NOTOP depth 1 superstable theory. In the other direction, we show that in the case of an -stable depth 2 theory, the L, (d.q)-theory is enough to describe the isomorphism type of all models.

• Type: Journal
• Year: 2006
• Where: APAL
• Authors: Elisabeth Bouscaren, Ehud Hrushovski
• Keywords: APAL 2006, Isomorphism Type, Structure Theory, Theory