Geodesic
On the absoluteness of $\aleph_1$-freeness
Algebra i logika, Tome 61 (2022) no. 5, pp. 523-540.

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$\aleph_1$-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.
Keywords: $\aleph_1$-free group, Pontryagin's criterion, absoluteness, transitive model. 
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D. Herden; A. V. Pasi. On the absoluteness of $\aleph_1$-freeness. Algebra i logika, Tome 61 (2022) no. 5, pp. 523-540. http://geodesic.mathdoc.fr/item/AL_2022_61_5_a0/

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