Torsion-Free Abelian Groups are Consistently $a{\Delta }_2^1$-complete

Abstract

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

21 pages