Scott Spectral Gaps are Bounded for Linear Orderings

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 $\alpha$.