  For a set $X\subseteq \mathbb{N}$, we define the $X$-torsion of a group $G$
to be all elements $g\in G$ with $g^{n}=e$ for some $n\in X$. With $X$
recursively enumerable, we give two independent proofs (group-theoretic, and
model-theoretic) that there exists a universal finitely presented
$X$-torsion-free group; one which contains all finitely presented
$X$-torsion-free groups. We also show that, if $X$ is recursively enumerable,
then the set of finite presentations of $X$-torsion-free groups is
$\Pi_{2}^{0}$-complete in Kleene's arithmetic hierarchy.
