The complexity of Scott sentences of scattered linear orders

TL;DR

This paper investigates the logical complexity of Scott sentences for countable scattered linear orders, establishing tight bounds on their descriptive complexity based on Hausdorff rank, and demonstrating the completeness of certain classes.

Contribution

It proves that scattered linear orders of Hausdorff rank α have a Scott sentence of complexity $d$-$\Sigma_{2\alpha+1}$ and that this bound is optimal, also showing the class's complexity is $\Sigma_{2\alpha+2}$-complete.

Findings

Scott sentences for these orders have tight complexity bounds.

The optimal complexity for these Scott sentences is $d$-$\Sigma_{2\alpha+1}$.

The class of linear orders with a fixed Hausdorff rank is $\Sigma_{2\alpha+2}$-complete.

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.