The tree property (session 2)
60 mins,
110.27 MB,
MP3
44100 Hz,
250.92 kbits/sec
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About this item
| Description: |
Sinapova, D (University of Illinois at Chicago)
Tuesday 25 August 2015, 11:30-12:30 |
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| Created: | 2015-09-01 10:42 |
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| Collection: | Mathematical, Foundational and Computational Aspects of the Higher Infinite |
| Publisher: | Isaac Newton Institute |
| Copyright: | Sinapova, D |
| 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: | The tree propperty at κ says that every tree of height κ and levels of size less than κ has a cofinal branch. A long term project in set theory is to get the consistency of the tree property at every regular cardinal greater than ℵ1. So far we only know that it is possible to have the tree property up to ℵω+1, due to Neeman. The next big hurdle is to obtain it both at ℵω+1 and ℵomega+2 when ℵω is trong limit. Doing so would require violating the singular cardinal hypothesis at ℵomega. In this tutorial we will start with some classic facts about the tree property, focusing on branch lemmas, successors of singulars and Prikry type forcing used to negate SCH. We will then go over recent developments including a dichotomy theorem about which forcing posets are good candidates for getting the tree property at ℵω+1 together with not SCH at ℵω. Finally, we will discuss the problem of obtaining the tree property at the first and double successors of a singular cardinal simultaneously. |
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