Characterizing the existence of a Borel complete expansion

TL;DR

This paper develops a framework to analyze potential canonical Scott sentences and characterizes when a theory has a Borel complete expansion, linking it to automorphism groups of models and providing results on theories with bounded classes.

Contribution

It introduces a machinery to relate canonical Scott sentences to structures and characterizes Borel completeness via automorphism groups, with new results on theories with bounded equivalence classes.

Findings

Borel completeness characterized by automorphism groups dividing $S__$ for some model.

Theories with bounded cross-cutting equivalence classes are not Borel complete.

Provides a converse to a previous theorem on Borel completeness.

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}.