Constructing strongly equivalent nonisomorphic models for unstable theories

Abstract

If T is an unstable theory of cardinality <λ or countable stable theory with OTOP or countable superstable theory with DOP, λ ω λ ω 1 in the superstable with DOP case) is regular and λ < λ = λ, then we construct for T strongly equivalent nonisomorphic models of cardinality λ. This can be viewed as a strong nonstructure theorem for such theories. We also consider the case when T is unsuperstable and develop further a result of Shelah about the existence of L ∞,λ-equivalent nonisomorphic models for such T. In addition, we show that a natural analogue of Scott's isomorphism theorem fails for models of power κ, if κ ω is regular, assuming κ < κ = κ.