Abstract:$\aleph_1$-free groups, Abelian groups for which every countable subgroup is free, exhibit a number of interesting algebraic and set-theoretic properties. We will 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 $\mathbf{M}$ 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 a starting point for further exploring the relationship between the set-theoretic and algebraic properties of $\aleph_1$-free groups. In particular, we will demonstrate how proofs may be dramatically simplified using model extensions for $\aleph_1$-free groups.
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