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**1 - 2**of**2**### Models with High Scott Rank

, 2008

"... 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 ..."

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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

### INVARIANT MEASURES CONCENTRATED ON COUNTABLE STRUCTURES

"... Abstract. Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the ..."

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Abstract. Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraïssé limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit