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- W2964341356 abstract "The Scott rank of a countable structure is a measure, coming from the proof of Scott's isomorphism theorem, of the complexity of that structure. The Scott spectrum of a theory (by which we mean a sentence of L ω 1 ω ) is the set of Scott ranks of countable models of that theory. In Z F C + P D we give a descriptive-set-theoretic classification of the sets of ordinals which are the Scott spectrum of a theory: they are particular Σ 1 1 classes of ordinals. Our investigation of Scott spectra leads to the resolution (in ZFC ) of a number of open problems about Scott ranks. We answer a question of Montalbán by showing, for each α < ω 1 , that there is a Π 2 in theory with no models of Scott rank less than α . We also answer a question of Knight and Calvert by showing that there are computable models of high Scott rank which are not computably approximable by models of low Scott rank. Finally, we answer a question of Sacks and Marker by showing that δ 2 1 is the least ordinal α such that if the models of a computable theory T have Scott rank bounded below ω 1 , then their Scott ranks are bounded below α ." @default.
- W2964341356 created "2019-07-30" @default.
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- W2964341356 date "2018-05-01" @default.
- W2964341356 modified "2023-09-24" @default.
- W2964341356 title "Scott ranks of models of a theory" @default.
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- W2964341356 doi "https://doi.org/10.1016/j.aim.2018.03.012" @default.
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