Decidability and computability of certain torsion-free abelian groups (Q976537)

The paper investigates the computable properties of torsion-free abelian groups of the form \({\mathcal G}_S = \bigoplus_{n \in S} {\mathbb{Q}}_{p_n}\), for sets \(S \subseteq \omega\), where \(p_n\) is the \(n\)-th prime number and \({\mathbb{Q}}_p\) is the subgroup of \(({\mathbb{Q}}, +)\) generated by the numbers \(1/p^k\), for \(k \in \omega\). It is shown that \({\mathcal G}_S\) has a decidable copy if and only if \(S\) is \(\Sigma_2^0\) and has a computable copy if and only if \(S\) is \(\Sigma_3^0\).