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

TL;DR

This paper demonstrates that the theory of torsion-free abelian groups is consistently $a \Delta^1_2$-complete under certain set-theoretic assumptions, and explores properties related to the Schr"{o}der-Bernstein property within this context.

Contribution

It establishes the $a \Delta^1_2$-completeness of TFAG under specific set-theoretic conditions and introduces the $oldsymbol{ ext{Schr"{o}der-Bernstein}}$ property for this theory.

Findings

TFAG is $a \\Delta^1_2$-complete if no countable transitive model of $ZFC^- + \\kappa(\\ ext{omega})$ exists.

TFAG fails the $oldsymbol{ ext{alpha-ary Schr"{o}der-Bernstein}}$ property for all $oldsymbol{ ext{alpha} < \\\kappa( ext{omega})}$.

Open question on whether TFAG has the $oldsymbol{ ext{kappa}( ext{omega})}$-ary Schr"{o}der-Bernstein property.

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.