Decidability and Computability of Certain Torsion-Free Abelian Groups

Abstract

We study completely decomposable torsion-free abelian groups of the form ${\mathcal{G}}_S:={\oplus }_{n\in S}{\mathbb{Q}}_{p_n}$ for sets $S\subseteq \omega$. We show that ${\mathcal{G}}_S$has a decidable copy if and only if S is ${\mathrm{\Sigma }}_2^0$and has a computable copy if and only if S is ${\mathrm{\Sigma }}_3^0$.