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University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > The tree property (session 1)

## The tree property (session 1)Add to your list(s) Download to your calendar using vCal - Sinapova, D (University of Illinois at Chicago)
- Monday 24 August 2015, 11:30-12:30
- 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 The tree propperty at $kappa$ says that every tree of height $kappa$ and levels of size less than $kappa$ 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 $leph_1$. So far we only know that it is possible to have the tree property up to $leph_{omega+1}$, due to Neeman. The next big hurdle is to obtain it both at $leph_{omega+1}$ and $leph_{omega+2}$ when $leph_omega$ is trong limit. Doing so would require violating the singular cardinal hypothesis at $leph_omega$. 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 $leph_{omega+1}$ together with not SCH at $leph_omega$. Finally, we will discuss the problem of obtaining the tree property at the first and double successors of a singular cardinal simultaneously. 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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