On the absoluteness of $\aleph_1$-freeness

Abstract

$\aleph_1$-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.