On the Absoluteness of ℵ1-Freeness

ℵ₁-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 ℵ₁-free is absolute; that is, if an Abelian group G is ℵ₁-free in some transitive model M of ZFC, then it is ℵ₁-free in any transitive model of ZFC containing G. The absoluteness of ℵ₁-freeness has the following remarkable consequence: an Abelian group G is ℵ₁-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 ℵ₁-free groups. In particular, we will demonstrate how proofs may be dramatically simplified using model extensions for ℵ₁-free groups.