April 12
Rachael Alvir, University of Notre Dame
Scott Sentences of Scattered Linear Orders

The logic $L_{\omega_1 \omega}$ is obtained by closing finitary first-order logic under countable disjunction and conjunction. There is a kind of normal form for such sentences. For any structure $\mathcal{A}$ there is a sentence of $L_{\omega_1 \omega}$, known as its Scott sentence, which describes $\mathcal{A}$ up to isomorphism among countable structures. Given a countable scattered linear order $L$ of Hausdorff rank $\alpha < \omega_1$, we show that it has a $d$-$\Sigma_{2 \alpha+1}$ Scott sentence. From Ash's calculation of the back and forth relations for all countable well-orders, 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.