Scott Ranks of Models of a Theory

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 Lω1ω) 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 Σ1 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 < ω1, that there is a Π 2 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 is the least ordinal α such that if the models of a computable theory T have Scott rank bounded below ω1, then their Scott ranks are bounded below α.