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Forcing, regularity properties and the axiom of choice

Tuesday 25th August 2015 - 14:00 to 14:30
INI Seminar Room 1
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 $ZF+\neg AC_{\omega}$, 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.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons