Borel Complexity and the Schröder-Bernstein Property

Abstract

We introduce a new invariant of Borel reducibility, namely the notion of thickness; this associates to every sentence $\hat{I}¦$ of ${\mathcal{L}}_{\ddot{I}{‰}_1\ddot{I}‰}$ and to every cardinal $\hat{I}»$, the thickness $\ddot{I}„(\hat{I}¦,\hat{I}»)$ of $\hat{I}¦$ at $\hat{I}»$. As applications, we show that all the Friedman-Stanley jumps of torsion abelian groups are non-Borel complete. We also show that under the existence of large cardinals, if $\hat{I}¦$ is a sentence of ${\mathcal{L}}_{\ddot{I}{‰}_1\ddot{I}‰}$ with the Schröder-Bernstein property (that is, whenever two countable models of $\hat{I}¦$ are biembeddable, then they are isomorphic), then $\hat{I}¦$ is not Borel complete.