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• W1831390059 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 $mathcal{L}_{omega_1 omega}$) is the set of Scott ranks of countable models of that theory. In $ZFC + PD$ we give a descriptive-set-theoretic classification of the sets of ordinals which are the Scott spectrum of a theory: they are particular $boldsymbol{Sigma}^1_1$ 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 Montalb'an by showing, for each $alpha < omega_1$, that there is a $Pi^{mathtt{in}}_2$ theory with no models of Scott rank less than $alpha$. 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 $delta^1_2$ is the least ordinal $alpha$ such that if the models of a computable theory $T$ have Scott rank bounded below $omega_1$, then their Scott ranks are bounded below $alpha$." @default.
• W1831390059 created "2016-06-24" @default.
• W1831390059 creator A5027707580 @default.
• W1831390059 date "2015-10-27" @default.
• W1831390059 modified "2023-09-27" @default.
• W1831390059 title "Scott ranks of models of a theory" @default.
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