Scott Spectral Gaps are Bounded for Linear Orderings

TL;DR

This paper proves that for linear orderings, the complexity gap between a theory and its models' Scott ranks is always bounded by four, contrasting with more complex structures.

Contribution

It establishes a uniform bound on the Scott rank gap for models of linear orderings and provides new lower bounds demonstrating the limits of this bound.

Findings

Bounded Scott rank gap of 4 for linear orderings.

Existence of theories with models having Scott ranks close to the theory complexity.

New lower bounds on Scott ranks for certain linear ordering theories.

Abstract

We demonstrate that any $\Pi_\alpha$ sentence of the infinitary logic $L_{\omega_1 \omega}$ extending the theory of linear orderings has a model with a $\Pi_{\alpha+4}$ Scott sentence and hence of Scott rank at most $\alpha+3$. In other words, the gap between the complexity of the theory and the complexity of the simplest model is always bounded by $4$. This contrasts the situation with general structures where for any $\alpha$ there is a $\Pi_2$ sentence all of whose models have Scott rank $\alpha$. We also give new lower bounds, though there remains a small gap between our lower and upper bounds: For most (but not all) $\alpha$, we construct a $\Pi_\alpha$ sentence extending the theory of linear orderings such that no models have a $\Sigma_{\alpha+2}$ Scott sentence and hence no models have Scott rank less than or equal to $\alpha$.