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SUMMARY:Forcing\, regularity properties and the axiom of choice - Horowitz
 \, H (Hebrew University of Jerusalem)
DTSTART:20150825T130000Z
DTEND:20150825T133000Z
UID:TALK60453@talks.cam.ac.uk
CONTACT:42080
DESCRIPTION:We consider general regularity properties associated with Susl
 in ccc forcing notions. By Solovay's celebrated work\, starting from a mod
 el of $ZFC+$"There exists an inaccessible cardinal"\, we can get a model o
 f $ZF+DC+$"All sets of reals are Lebesgue measurable and have the Baire pr
 operty". By another famous result of Shelah\, $ZF+DC+$"All sets of reals h
 ave the Baire property" is equiconsistent with $ZFC$. This result was obta
 ined by isolating the notion of "sweetness"\, a strong version of ccc whic
 h is preserved under amalgamation\, thus allowing the construction of a su
 itably homogeneous forcing notion.\n\nThe above results lead to the follow
 ing question: Can we get a similar result for non-sweet ccc forcing notion
 s without using an inaccessible cardinal?\n\nIn our work we give a positiv
 e answer by constructing a suitable ccc creature forcing and iterating alo
 ng a non-wellfounded homogeneous linear order. While the resulting model s
 atisfies $ZF+\neg AC_{omega}$\, we prove in a subsequent work that startin
 g with a model of $ZFC+$"There is a measurable cardinal"\, we can get a mo
 del of $ZF+DC_{omega_1}$. This is joint work with Saharon Shelah.\n
LOCATION:Seminar Room 1\, Newton Institute
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