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SUMMARY:Indestructible remarkable cardinals - Gitman\, V (City University 
 of New York)
DTSTART:20150828T150000Z
DTEND:20150828T160000Z
UID:TALK60501@talks.cam.ac.uk
CONTACT:42080
DESCRIPTION:In 2000\, Schindler introduced remarkable cardinals and showed
  that the existence of a remarkable cardinal is equiconsistent with the as
 sertion that the theory of $L(mathbb R)$ is absolute for proper forcing. R
 emarkable cardinals can be thought of either as a  miniature version of st
 rong cardinals or as having aspects of generic supercompactness\, but they
  are relatively low in the large cardinal hierarchy. They are downward abs
 olute to $L$ and lie (consistency-wise) between the 1-iterable and 2-itera
 ble cardinals of the $lpha$-iterable cardinals hierarchy (below Ramsey ca
 rdinals). I will discuss the indestructibility properties of remarkable ca
 rdinals\, which are similar to those of strong cardinals. I will show that
  a remarkable cardinal $kappa$ can be made simultaneously indestructible b
 y all $ltkappa$-closed $leqkappa$-distributive forcing and by all forcing 
 of the form ${\nm Add}(kappa\,	heta)*mathbb R$\, where $mathbb R$ is force
 d to be $ltkappa$-closed and $leqkappa$-distributive. For this argument\, 
 I will introduce the notion of a remarkable Laver function and show that e
 very remarkable cardinal has one. Although\, the existence of Laver-like f
 unctions can be forced for most large cardinals\, few\, such as strong\, s
 upercompact\, and extendible cardinals\, have them outright. The establish
 ed indestructibility can be used to show\, for instance\, that any consist
 ent continuum pattern on the regular cardinals can be realized above a rem
 arkable cardinal and that a remarkable cardinal need not be even weakly co
 mpact in ${\nm HOD}$. This is joint work with Yong Cheng.\n
LOCATION:Seminar Room 1\, Newton Institute
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