Torsion-free abelian groups are consistently $ {\rm a}\Delta ^1_2$-complete

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Let $\mbox{TFAG}$ be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of $\mbox{ZFC}^- + \text{“}\kappa(\omega)$ exists”, then $\mbox{TFAG}$ is $\mathrm{a}\Delta^1_2$-complete; in particular, this is consis