Scott ranks of models of a theory

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 $\mathcal{L}_{ω_1 ω}$) 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 $\boldsymbolΣ^1_1$ 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 $Π^{\mathtt{in}}_2$ 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 $δ^1_2$ 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 $α$.