Although written in the context of model theory, this paper should be of interest to abelian group theorists, since the main results state that the theory of finitely valued p-valuated groups whose underlying group is the countable direct sum of copies of \({\mathbb{Z}}(p^ 9)\) is hereditarily undecidable, and so is the theory of countable valued p-valuated countable p-local torsion-free abelian groups. One consequence of these results is that it is probably hopeless to seek a complete classification of these classes. The proof is based on the fact that the theory of countable \(p^ 9\)- bounded groups with a distinguished subgroup is reducible to the theory of valuated groups; but the former theory was proved hereditarily undecidable by \textit{W. Baur} [Proc. Am. Math. Soc. 55, 125-128 (1976; Zbl 0328.02032)].