Forcing ℵ1-free groups to be free.
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Pasi, Alexandra V., 1993-
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ℵ1-free groups, abelian groups whose countable subgroups are free, are objects of both algebraic and set-theoretic interest. Illustrating this, we note that ℵ1-free groups, and in particular the question of when ℵ1-free groups are free, were central to the resolution of the Whitehead problem as undecidable. In elucidating the relationship between ℵ1-freeness and freeness, we prove the following result: an abelian group G is ℵ1-free in a countable transitive model of ZFC (and thus by absoluteness, in every transitive model of ZFC) if and only if it is free in some generic model extension. We would like to answer the more specific question of when an ℵ1-free group can be forced to be free while preserving the cardinality of the group. For groups of size ℵ1, we establish a necessary and sufficient condition for when such forcings are possible. We also identify both existing and novel forcings which force such ℵ1-free groups of size ℵ1 to become free with cardinal preservation. These forcings lay the groundwork for a larger project which uses forcing to explore various algebraic properties of ℵ1-free groups and develops new set-theoretical tools for working with them.