Torsion-free abelian groups are consistently aΔ12-complete

Abstract

Let TFAG be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of ZFC- + "R(ω) exists", then TFAG is aΔ12- complete; in particular, this is consistent with ZFC. We define the α-ary Schröder- Bernstein property, and show that TFAG fails the α-ary Schröder-Bernstein property for every α < R (ω). We leave open whether or not TFAG can have the R(ω)-ary Schröder- Bernstein property; if it did, then it would not be aΔ12 -complete, and hence not Borel complete.