Scott ranks of models of a theory

TL;DR

This paper classifies the possible sets of Scott ranks of countable models of theories using descriptive set theory, and resolves several open questions about Scott ranks in computability and model theory.

Contribution

It provides a descriptive-set-theoretic classification of Scott spectra and answers multiple open problems regarding Scott ranks and computable models.

Findings

Scott spectra are particular $oldsymbol{oldsymbol{ ext{}}}oldsymbol{oldsymbol{ ext{}}}oldsymbol{oldsymbol{ ext{}}}oldsymbol{oldsymbol{ ext{}}}oldsymbol{oldsymbol{ ext{}}}oldsymbol{oldsymbol{ ext{}}}oldsymbol{oldsymbol{ ext{}}}oldsymbol{ ext{}}$ classes of ordinals.

Existence of theories with models of arbitrarily high Scott rank below $ ext{omega}_1$.

Identification of the least $oldsymbol{ ext{delta}}^1_2$ ordinal bounding Scott ranks of models.

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 $\alpha$.