Scott Spectral Gaps are Bounded for Linear Orderings

Abstract

We demonstrate that any $Π_α$ sentence of the infinitary logic $L_{ω_1 ω}$ extending the theory of linear orderings has a model with a $Π_{α+4}$ Scott sentence and hence of Scott rank at most $α+3$. 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 $α$ there is a $Π_2$ sentence all of whose models have Scott rank $α$. We also give new lower bounds, though there remains a small gap between our lower and upper bounds: For most (but not all) $α$, we construct a $Π_α$ sentence extending the theory of linear orderings such that no models have a $Σ_{α+2}$ Scott sentence and hence no models have Scott rank less than or equal to $α$.