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arXiv · 2109.06140

Characterizing the existence of a Borel complete expansion

Michael C. Laskowski·Douglas S. Ulrich
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Abstract

We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence $\Phi$ as a class of structures in a related language. From this, we show that $\Phi$ has a Borel complete expansion if and only if $S_\infty$ divides $Aut(M)$ for some countable model $M\models \Phi$. Using this, we prove that for theories $T_h$ asserting that $\{E_n\}$ is a countable family of cross cutting equivalence relations with $h(n)$ classes, if $h(n)$ is uniformly bounded then $T_h$ is not Borel complete, providing a converse to Theorem~2.1 of \cite{LU}.

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Michael C. Laskowski, Douglas S. Ulrich. 2021-09-13. Characterizing the existence of a Borel complete expansion. https://arxiv.org/abs/2109.06140

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