Torsion-Free Abelian Groups are Consistently $a Δ^1_2$-complete

Abstract: Let $\mbox{TFAG}$ be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of $ZFC^- + \kappa(\omega)$ exists, then $\mbox{TFAG}$ is $a \Delta^1_2$-complete; in particular, this is consistent with $ZFC$. We define the $\alpha$-ary Schr\"{o}der- Bernstein property, and show that $\mbox{TFAG}$ fails the $\alpha$-ary Schr\"{o}der-Bernstein property for every $\alpha < \kappa(\omega)$. We leave open whether or not $\mbox{TFAG}$ can have the $\kappa(\omega)$-ary Schr\"{o}der-Bernstein property; if it did, then it would not be $a \Delta^1_2$-complete, and hence not Borel complete.