Scott Spectral Gaps are Bounded for Linear Orderings

Abstract

We demonstrate that any ${\hat{I}}_{\hat{I}}\pm$ sentence of the infinitary logic $L_{\ddot{I}{‰}_1\ddot{I}‰}$ extending the theory of linear orderings has a model with a ${\hat{I}}_{\hat{I}\pm +4}$ Scott sentence and hence of Scott rank at most $\hat{I}\pm +3$. 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 $\hat{I}\pm$ there is a ${\hat{I}}_2$ sentence all of whose models have Scott rank $\hat{I}\pm$. We also give new lower bounds, though there remains a small gap between our lower and upper bounds: For most (but not all) $\hat{I}\pm$, we construct a ${\hat{I}}_{\hat{I}}\pm$ sentence extending the theory of linear orderings such that no models have a $\hat{I}{\pounds }_{\hat{I}\pm +2}$ Scott sentence and hence no models have Scott rank less than or equal to $\hat{I}\pm$.

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