Forcing, regularity properties and the axiom of choice
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About this item
| Description: |
Horowitz, H (Hebrew University of Jerusalem)
Tuesday 25 August 2015, 14:00-14:30 |
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| Created: | 2015-09-01 09:34 |
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| Collection: | Mathematical, Foundational and Computational Aspects of the Higher Infinite |
| Publisher: | Isaac Newton Institute |
| Copyright: | Horowitz, H |
| Language: | eng (English) |
| Distribution: |
World
(downloadable)
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| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| Abstract: | 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+¬ACω, 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+DCω1. This is joint work with Saharon Shelah. |
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