The main results of this paper are the following: For any family \(R\) of finite sets there exists a completely decomposable torsion-free abelian group \(G_{R}\) of infinite rank such that \(G_{R}\) has an \(X\)-computable copy if and only if \(R\) has a \(\Sigma_{2}^{X}\)-computable enumeration (Theorem 4). There exists a completely decomposable torsion-free abelian group \(G\) of infinite rank such that \(G\) has an \(X\)-decomposable copy if and only if \(X^{\prime}>_{T}0^{\prime}\) (Theorem 5). As a consequence of these results, it is proved that there exists a completely decomposable torsion-free abelian group \(G\) of infinite rank with no jump degree.