Abstract

Scott rank is a measure of model-theoretic complexity; the Scott rank of a structure A in the language L is the least ordinal β for which A is prime in its Lωβ,ω-theory. By a result of Nadel, the Scott rank of a structure A is at most ωA 1 + 1, where ωA 1 is the least ordinal not recursive in A. We say that the Scott rank of A is high if it is at least ωA 1. Let α be a Σ1 admissible ordinal. A structure A of high Scott rank (and for which ω A 1 = α) will have Scott rank α + 1 if it realizes a non-principal Lα,ω-type, and Scott rank α otherwise. For α = ω CK 1, the least non-recursive ordinal, several sorts of constructions are known. The Harrison ordering ω CK 1 (1 + η), where η is the order-type of the rationals, has Scott rank ω CK 1 + 1. Makkai constructs a model with Scott rank ω CK 1 whose L ω CK 1,ω-theory is ℵ0-categorical. Millar and Sacks produce a model A with Scott rank ω CK 1 (in which ω A 1 = ω CK 1) but whose L ω CK 1,ω-theory is not

Citations