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The tree property (session 1)

Monday 24th August 2015 - 11:30 to 12:30
INI Seminar Room 1
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 $\aleph_1$. So far we only know that it is possible to have the tree property up to $\aleph_{\omega+1}$, due to Neeman. The next big hurdle is to obtain it both at $\aleph_{\omega+1}$ and $\aleph_{omega+2}$ when $\aleph_\omega$ is trong limit. Doing so would require violating the singular cardinal hypothesis at $\aleph_omega$. 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 $\aleph_{\omega+1}$ together with not SCH at $\aleph_\omega$. 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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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons