The complexity of Scott sentences of scattered linear orders

Abstract

Given a countable scattered linear order $L$ of Hausdorff rank $\alpha < \omega_1$ we show that it has a $d\text{-}\Sigma_{2\alpha+1}$ Scott sentence. Ash calculated the back and forth relations for all countable well-orders. From this result we obtain that this upper bound is tight, i.e., for every $\alpha < \omega_1$ there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable $\alpha$ the class of Hausdorff rank $\alpha$ linear orders is $\pmb \Sigma_{2\alpha+2}$ complete.