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arXiv · 1810.11423

The complexity of Scott sentences of scattered linear orders

Rachael Alvir·Dino Rossegger
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Abstract

Given a countable scattered linear order $L$ of Hausdorff rank $α< ω_1$ we show that it has a $d\text{-}Σ_{2α+1}$ 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 $α< ω_1$ there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable $α$ the class of Hausdorff rank $α$ linear orders is $\pmb Σ_{2α+2}$ complete.

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Rachael Alvir, Dino Rossegger. 2019-09-16. The complexity of Scott sentences of scattered linear orders. https://doi.org/10.1017/jsl.2020.46

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