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SUMMARY:Ultrafilters without p-point quotients - Goldstern\, M (Technische
  Universitt Wien)
DTSTART:20150828T090000Z
DTEND:20150828T100000Z
UID:TALK60497@talks.cam.ac.uk
CONTACT:42080
DESCRIPTION:A p-point is a nonprincipal ultrafilter on the set N of natura
 l numbers \nwhich has the property that for every countable family of filt
 er sets\nthere is a pseudointersection in the filter\, i.e. a filter set w
 hich \nis almost contained in each set of the family.  Equivalently\, a\np
 -point is an element of the Stone-Cech remainder beta(N) minus N\nwhose ne
 ighborhood filter is closed under countable intersections. \n\nIt is well 
 known that p-points "survive" various forcing iterations\,\nthat is: exten
 ding a universe V with certain forcing iterations P\nwill result in a univ
 erse V' in which all (or at least: certain \nwell-chosen) p-points are sti
 ll ultrafilter bases in the extension. \nThis shows that the sentence "The
  continuum hypothesis is false\,\nyet there are aleph1-generated ultrafilt
 ers\, namely: certain \np-points" is relatively consistent with ZFC.  \n\n
 In a joint paper with Diego Mejia and Saharon Shelah (still in\nprogress) 
 we construct ultrafilters on N which are\, on the \none hand\, far away fr
 om being p-points (there is no Rudin-Keisler\nquotient which is a p-point)
 \, but on the other hand can \nsurvive certain forcing iterations adding r
 eals but killing\np-points.  This shows that non-CH is consistent with \ns
 mall ultrafilter bases AND the nonexistence of p-points. \n
LOCATION:Seminar Room 1\, Newton Institute
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