### Torsion-free abelian groups are consistently $a\Delta\frac{1}{2}$-complete

by Shelah and Ulrich. [ShUl:1141]

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

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