If T is an unstable theory of cardinality <λ or countable stable theory with OTOP or countable superstable theory with DOP, λω λω1 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 κω is regular, assuming κ<κ=κ. (shrink)

Let σ(U) denote the number of automorphisms of a model U of power ω1. We derive a necessary and sufficient condition in terms of trees for the existence of an U with $\omega_1 < \sigma(\mathfrak{U}) < 2^{\omega_1}$. We study the sufficiency of some conditions for σ(U) = 2ω1 . These conditions are analogous to conditions studied by D. Kueker in connection with countable models.

Let $\sigma$ denote the number of automorphisms of a model $\mathfrak{U}$ of power $\omega_1$. We derive a necessary and sufficient condition in terms of trees for the existence of an $\mathfrak{U}$ with $\omega_1 < \sigma < 2^{\omega_1}$. We study the sufficiency of some conditions for $\sigma = 2^{\omega_1}$. These conditions are analogous to conditions studied by D. Kueker in connection with countable models.