The decomposability problem for torsion-free abelian groups is analytic complete

Abstract

We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is $\Sigma^0_3$-complete. However, when we consider groups of infinite rank, it becomes $\Sigma^1_1$-complete, so it cannot be characterized by a first-order formula in the language of arithmetic.