# Computability Learning Seminar

Tuesday, September 17, 2019 — 2:00 PM EDT

Dino Rossegger, Department of Pure Mathematics, University of Waterloo

"The complexity of Scott sentences of scattered linear orders"

Scott showed that for every countable structure $\mathcal A$ there is a sentence $\phi$ in the infinitary logic $L_{\omega_1\omega}$ such that for all countable structures $\mathcal B$, $\mathcal B\cong \mathcal A$ if and only if $\mathcal B \models \phi$. This sentence is commonly known as the Scott sentence of $\mathcal A$. Similarly to first order logic we can measure the complexity of sentences in $L_{\omega_1\omega}$ and obtain natural complexity classes $\Sigma_\alpha$, $\Pi_\alpha$ and $d\text{-}\Sigma_\alpha$ for all countable ordinals $\alpha$. The complexity of a structures least complex Scott sentence is strongly connected to its Scott rank and thus provides a good measure of complexity of a structure.

We investigate the complexity of Scott sentences of \emph{scattered} linear orders -- linear orders which do not have dense suborders. Hausdorff gave an inductive definition of the countable scattered linear orders. This allows us to assign an ordinal to every countable scattered linear order, its \emph{Hausdorff rank}. We obtain tight bounds on the complexity of the Scott sentences of countable scattered linear orders.

Theorem 1. Let $L$ be a linear order with countable Hausdorff rank $\alpha$. Then it has a $d\text{-}\Sigma_{2\alpha+1}$ Scott sentence.

Our result is optimal in the sense that for every countable $\alpha$ there is a linear order of Hausdorff rank $\alpha$ which does not have a Scott sentence of less complexity. This is a drastic improvement of bounds obtained by Nadel  and extends work of Ash who calculated the Scott rank of countable well orders.

We furthermore show that for all countable $\alpha$, the set of presentations of linear orders of Hausdorff rank $\alpha$ is $\pmb \Sigma_{2\alpha+2}$ complete and that for $\alpha<\omega_1^{\mathrm CK}$, the index set of linear orders of Hausdorff rank $\alpha$ is $\Sigma_{2\alpha+2}$ complete.

This is joint work with Rachael Alvir, University of Notre Dame.

MC 5413

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