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SUMMARY:The tree property (session 1) - Sinapova\, D (University of Illino
 is at Chicago)
DTSTART:20150824T103000Z
DTEND:20150824T113000Z
UID:TALK60435@talks.cam.ac.uk
CONTACT:42080
DESCRIPTION: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 t
 erm project in set theory is to get the consistency of the tree property a
 t 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 requir
 e violating the singular cardinal hypothesis at $leph_omega$. \n\nIn this
  tutorial we will start with some classic facts about the tree property\, 
 focusing on branch lemmas\, successors of singulars and Prikry type forcin
 g used to negate SCH. We will then go over recent developments including a
  dichotomy theorem about which forcing posets are good candidates for gett
 ing the tree property at $leph_{omega+1}$ together with not SCH at $leph
 _omega$. Finally\, we will discuss the problem of obtaining the tree prope
 rty at the first and double successors of a singular cardinal simultaneous
 ly.\n
LOCATION:Seminar Room 1\, Newton Institute
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