Models with high Scott rank

Scott rank is a measure of model-theoretic complexity; the Scott rank of a structure [Special characters omitted.] in the language [Special characters omitted.] is the least ordinal β for which [Special characters omitted.] is prime in its [Special characters omitted.] -theory. By a result of Nadel, the Scott rank of a structure [Special characters omitted.] is at most [Special characters omitted.] + 1, where [Special characters omitted.] is the least ordinal not recursive in [Special characters omitted.] . We say that the Scott rank of [Special characters omitted.] is high if it is at least [Special characters omitted.] . Let α be a ∑ 1 admissible ordinal. A structure [Special characters omitted.] of high Scott rank (and for which [Special characters omitted.] = α) will have Scott rank α + 1 if it realizes a non-principal [Special characters omitted.] -type, and Scott rank α otherwise. For α = [Special characters omitted.] , the least non-recursive ordinal, several sorts of constructions are known. The Harrison ordering [Special characters omitted.] (1 + η), where η is the order-type of the rationals, has Scott rank [Special characters omitted.] + 1. Makkai constructs a model with Scott rank [Special characters omitted.] whose [Special characters omitted.] -theory is [Special characters omitted.] -categorical. Millar and Sacks produce a model [Special characters omitted.] with Scott rank [Special characters omitted.] (in which [Special characters omitted.] ) but whose [Special characters omitted.] -theory is not [Special characters omitted.] -categorical. We extend the result of Millar and Sacks to an arbitrary countable ∑ 1 admissible ordinal α. For such α, we show that there is a model [Special characters omitted.] with Scott rank α (in which [Special characters omitted.] = α) whose [Special characters omitted.] -theory is not [Special characters omitted.] -categorical. When α is a ∑ 1 admissible ordinal with ω 1 ≤ α < ω 2 we obtain a model with Scott rank α whose [Special characters omitted.] -theory is not [Special characters omitted.] -categorical, but we are unable to preserve the admissibility of α within this structure.