We calculate the complexity of Scott sentences of scattered linear orders. Given a countable scattered linear order L of Hausdorff rank $\alpha $ we show that it has a ${d\text {-}\Sigma _{2\alpha +1}}$ Scott sentence. It follows from results of Ash [2] that for every countable $\alpha $ there is a linear order whose optimal Scott sentence has this complexity. Therefore, our bounds are tight. We furthermore show that every Hausdorff rank 1 linear order has an optimal ${\Pi ^{\mathrm {c}}_{3}}$ or ${d\text {-}\Sigma ^{\mathrm {c}}_{3}}$ Scott sentence and give a characterization of those linear orders of rank $1$ with ${\Pi ^{\mathrm {c}}_{3}}$ optimal Scott sentences. At last we show that for all countable $\alpha $ the class of Hausdorff rank $\alpha $ linear orders is $\boldsymbol {\Sigma }_{2\alpha +2}$ complete and obtain analogous results for index sets of computable linear orders.

中文翻译：

---

散点线性顺序的 SCOTT 句子的复杂性

我们计算分散线性顺序的 Scott 句子的复杂度。给定一个可数的分散线性顺序大号豪斯多夫等级$\阿尔法$我们证明它有一个${d\text {-}\Sigma _{2\alpha +1}}$斯科特的句子。根据 Ash [2] 的结果，对于每个可数$\阿尔法$存在一个线性顺序，其最优 Scott 句子具有这种复杂性。因此，我们的界限是严格的。我们进一步表明，每个 Hausdorff 秩 1 线性阶都有一个最优${\Pi ^{\mathrm {c}}_{3}}$要么${d\text {-}\Sigma ^{\mathrm {c}}_{3}}$斯科特句子并给出这些线性等级顺序的表征$1$和${\Pi ^{\mathrm {c}}_{3}}$最佳斯科特句子。最后我们证明对于所有可数的$\阿尔法$豪斯多夫等级$\阿尔法$线性订单是$\boldsymbol {\Sigma }_{2\alpha +2}$完成并获得可计算线性阶的索引集的类似结果。