arXiv subjects

Douglas S. Ulrich

Publications and source records attributed to Douglas S. Ulrich.

3 recordsLinked to original sources

arXiv2023

Characterizing the existence of a Borel complete expansion

We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence $Φ$ as a class of structures in a related language. From this, we show that $Φ$ has a Borel complete expansion if and only if $S_\infty$ divides $Aut(M)$ for some countable model $M\models Φ$. 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}.

math.LO↗

arXiv2022

A proof of the Borel completeness of torsion free abelian groups

A proof of the Borel completeness of torsion free abelian groups is presented. This proof differs considerably from the approach of Paolini-Shelah.

math.LO↗

arXiv2021

Most(?) theories have Borel complete reducts

We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has a Borel complete reduct, and if a theory $T$ is not $ω$-stable, then the elementary diagram of some countable model of $T$ has a Borel complete reduct.

math.LO↗