  $\aleph_1$-free groups, abelian groups for which every countable subgroup is
free, exhibit a number of interesting algebraic and set-theoretic properties.
In this paper, we give a complete proof that the property of being
$\aleph_1$-free is absolute; that is, if an abelian group $G$ is
$\aleph_1$-free in some transitive model $\textbf{M}$ of ZFC, then it is
$\aleph_1$-free in any transitive model of ZFC containing $G$. The absoluteness
of $\aleph_1$-freeness has the following remarkable consequence: an abelian
group $G$ is $\aleph_1$-free in some transitive model of ZFC if and only if it
is (countable and) free in some model extension. This set-theoretic
characterization will be the starting point for further exploring the
relationship between the set-theoretic and algebraic properties of
$\aleph_1$-free groups. In particular, this paper will demonstrate how proofs
may be dramatically simplified using model extensions for $\aleph_1$-free
groups.
