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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Ultrafilters without p-point quotients - Goldstern
\, M (Technische Universitt Wien)
DTSTART;TZID=Europe/London:20150828T100000
DTEND;TZID=Europe/London:20150828T110000
UID:TALK60497AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/60497
DESCRIPTION:A p-point is a nonprincipal ultrafilter on the set
N of natural numbers \nwhich has the property tha
t for every countable family of filter sets\nthere
is a pseudointersection in the filter\, i.e. a fi
lter set which \nis almost contained in each set o
f the family. Equivalently\, a\np-point is an ele
ment of the Stone-Cech remainder beta(N) minus N\n
whose neighborhood filter is closed under countabl
e intersections. \n\nIt is well known that p-point
s "survive" various forcing iterations\,\nthat is:
extending a universe V with certain forcing itera
tions P\nwill result in a universe V' in which all
(or at least: certain \nwell-chosen) p-points are
still ultrafilter bases in the extension. \nThis
shows that the sentence "The continuum hypothesis
is false\,\nyet there are aleph1-generated ultrafi
lters\, namely: certain \np-points" is relatively
consistent with ZFC. \n\nIn a joint paper with Di
ego Mejia and Saharon Shelah (still in\nprogress)
we construct ultrafilters on N which are\, on the
\none hand\, far away from being p-points (there i
s no Rudin-Keisler\nquotient which is a p-point)\,
but on the other hand can \nsurvive certain forci
ng iterations adding reals but killing\np-points.
This shows that non-CH is consistent with \nsmall
ultrafilter bases AND the nonexistence of p-point
s. \n
LOCATION:Seminar Room 1\, Newton Institute
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