Torsion-free abelian groups are consistently $ {\rm a}\Delta ^1_2$-complete
Volume 247 / 2019
Fundamenta Mathematicae 247 (2019), 275-297
MSC: Primary 03C55.
DOI: 10.4064/fm673-12-2018
Published online: 12 August 2019
Abstract
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 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 $\alpha \lt \kappa(\omega)$. 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 $\mathrm{a} \Delta^1_2$-complete, and hence not Borel complete.
