  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.
