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University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Forcing, regularity properties and the axiom of choice

## Forcing, regularity properties and the axiom of choiceAdd to your list(s) Download to your calendar using vCal - Horowitz, H (Hebrew University of Jerusalem)
- Tuesday 25 August 2015, 14:00-14:30
- Seminar Room 1, Newton Institute.
If you have a question about this talk, please contact webseminars. Mathematical, Foundational and Computational Aspects of the Higher Infinite We consider general regularity properties associated with Suslin ccc forcing notions. By Solovay’s celebrated work, starting from a model of $ZFC+$”There exists an inaccessible cardinal”, we can get a model of $ZF+DC+$”All sets of reals are Lebesgue measurable and have the Baire property”. By another famous result of Shelah, $ZF+DC+$”All sets of reals have the Baire property” is equiconsistent with $ZFC$. This result was obtained by isolating the notion of “sweetness”, a strong version of ccc which is preserved under amalgamation, thus allowing the construction of a suitably homogeneous forcing notion. The above results lead to the following question: Can we get a similar result for non-sweet ccc forcing notions without using an inaccessible cardinal? In our work we give a positive answer by constructing a suitable ccc creature forcing and iterating along a non-wellfounded homogeneous linear order. While the resulting model satisfies $ZF+ eg AC_{omega}$, we prove in a subsequent work that starting with a model of $ZFC+$”There is a measurable cardinal”, we can get a model of $ZF+DC_{omega_1}$. This is joint work with Saharon Shelah. This talk is part of the Isaac Newton Institute Seminar Series series. ## This talk is included in these lists:- All CMS events
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