Forcing $\aleph_1$-Free Groups to Be Free

Abstract: $\aleph_1$-free groups, abelian groups whose countable subgroups are free, are objects of both algebraic and set-theoretic interest. Illustrating this, we note that $\aleph_1$-free groups, and in particular the question of when $\aleph_1$-free groups are free, were central to the resolution of the Whitehead problem as undecidable. In elucidating the relationship between $\aleph_1$-freeness and freeness, we prove the following result: an abelian group $G$ is $\aleph_1$-free in a countable transitive model of $\operatorname{ZFC}$ (and thus by absoluteness, in every transitive model of $\operatorname{ZFC}$) if and only if it is free in some generic model extension. We would like to answer the more specific question of when an $\aleph_1$-free group can be forced to be free while preserving the cardinality of the group. For groups of size $\aleph_1$, we establish a necessary and sufficient condition for when such forcings are possible. We also identify a number of existing and novel forcings which force such $\aleph_1$-free groups of size $\aleph_1$ to become free with cardinal preservation. These forcings lay the groundwork for a larger project which uses forcing to explore various algebraic properties of $\aleph_1$-free groups and develops new set-theoretical tools for working with them.

computational complexitycategory theorylogic

Audience: researchers in the topic

PALS Panglobal Algebra and Logic Seminar

Series comments: The PALS seminar is a research and learning seminar organized by the algebra and logic research group of the Department of Mathematics at the University of Colorado at Boulder. The scope of the seminar includes all topics with links to algebra, logic, or their applications, like general algebra, logic, model theory, category theory, set theory, set-theoretic topology, or theoretical computer science. Please contact one of the organizers for the Zoom password, to join the mailing list or if you want to speak.