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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:The tree property (session 3) - Sinapova\, D (Univ
ersity of Illinois at Chicago)
DTSTART;TZID=Europe/London:20150826T113000
DTEND;TZID=Europe/London:20150826T123000
UID:TALK60466AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/60466
DESCRIPTION:The tree propperty at $kappa$ says that every tree
of height $kappa$ and levels of size less than $k
appa$ 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 t
o have the tree property up to $leph_{omega+1}$\,
due to Neeman. The next big hurdle is to obtain i
t 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 $leph_omega$. \n\nIn this tutorial we will st
art with some classic facts about the tree propert
y\, focusing on branch lemmas\, successors of sing
ulars and Prikry type forcing used to negate SCH.
We will then go over recent developments including
a dichotomy theorem about which forcing posets ar
e 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 doubl
e successors of a singular cardinal simultaneously
.\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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