BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Lexi V. Pasi (Baylor University) DTSTART:20210309T200000Z DTEND:20210309T210000Z DTSTAMP:20260423T021729Z UID:PALS/4 DESCRIPTION:Title: For cing $\\aleph_1$-Free Groups to Be Free\nby Lexi V. Pasi (Baylor Unive rsity) as part of PALS Panglobal Algebra and Logic Seminar\n\n\nAbstract\n $\\aleph_1$-free groups\, abelian groups whose countable subgroups are fre e\, are objects of both algebraic and set-theoretic interest. Illustrating this\, we note that $\\aleph_1$-free groups\, and in particular the quest ion of when $\\aleph_1$-free groups are free\, were central to the resolut ion of the Whitehead problem as undecidable. In elucidating the relationsh ip between $\\aleph_1$-freeness and freeness\, we prove the following resu lt: an abelian group $G$ is $\\aleph_1$-free in a countable transitive mod el of $\\operatorname{ZFC}$ (and thus by absoluteness\, in every transitiv e model of $\\operatorname{ZFC}$) if and only if it is free in some generi c model extension. We would like to answer the more specific question of w hen an $\\aleph_1$-free group can be forced to be free while preserving th e 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 p reservation. These forcings lay the groundwork for a larger project which uses forcing to explore various algebraic properties of $\\aleph_1$-free g roups and develops new set-theoretical tools for working with them.\n LOCATION:https://researchseminars.org/talk/PALS/4/ END:VEVENT END:VCALENDAR