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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Forcing\, regularity properties and the axiom of c
hoice - Horowitz\, H (Hebrew University of Jerusal
em)
DTSTART;TZID=Europe/London:20150825T140000
DTEND;TZID=Europe/London:20150825T143000
UID:TALK60453AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/60453
DESCRIPTION:We consider general regularity properties associat
ed with Suslin ccc forcing notions. By Solovay's c
elebrated work\, starting from a model of $ZFC+$"T
here exists an inaccessible cardinal"\, we can get
a model of $ZF+DC+$"All sets of reals are Lebesgu
e measurable and have the Baire property". By anot
her famous result of Shelah\, $ZF+DC+$"All sets of
reals have the Baire property" is equiconsistent
with $ZFC$. This result was obtained by isolating
the notion of "sweetness"\, a strong version of cc
c which is preserved under amalgamation\, thus all
owing the construction of a suitably homogeneous f
orcing notion.\n\nThe above results lead to the fo
llowing question: Can we get a similar result for
non-sweet ccc forcing notions without using an ina
ccessible cardinal?\n\nIn our work we give a posit
ive answer by constructing a suitable ccc creature
forcing and iterating along a non-wellfounded hom
ogeneous linear order. While the resulting model s
atisfies $ZF+\neg AC_{omega}$\, we prove in a subs
equent work that starting with a model of $ZFC+$"T
here is a measurable cardinal"\, we can get a mode
l of $ZF+DC_{omega_1}$. This is joint work with Sa
haron Shelah.\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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