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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:The Search for Deep Inconsistency - Koellner\, P (
Harvard University)
DTSTART;TZID=Europe/London:20150825T090000
DTEND;TZID=Europe/London:20150825T100000
UID:TALK60450AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/60450
DESCRIPTION:The hierarchy of large cardinals provides us with
a canonical means to climb the hierarchy of consis
tency strength. There have been any purported inc
onsistency proofs of various large cardinal axioms
. For example\, there have been many proofs purpor
ting to show that measurable cardinals are inconsi
stent. But to date the only proofs that have stood
the test of time are those which are rather trans
parent and simple\, the most notable example being
Kunen's proof showing that Reinhardt cardinals ar
e inconsistent. The Kunen result\, however\, makes
use of AC. And long standing open question is whe
ther Reinhardt cardinals are consistent in the con
text of ZF.\n\nIn this talk I will survey the simp
le inconsistency proofs and then raise the questio
n of whether perhaps the large cardinal hierarchy
outstrips AC\, passing through Reinhardt cardinals
and reaching far beyond. There are two main motiv
ations for this investigation. First\, it is of in
terest in its own right to determine whether the h
ierarchy of consistency strength outstrips AC. Per
haps there is an entire "choicless" large cardinal
hierarchy\, one which reaches new consistency str
engths and has fruitful applications. Second\, sin
ce the task of proving an inconsistency result bec
omes easier as one strengthens the hypothesis\, in
the search for a deep inconsistency it is reasona
ble to start with outlandishly strong large cardin
al assumptions and then work ones way down. This w
ill lead to the formulation of large cardinal axio
ms (in the context of ZF) that start at the level
of a Reinhardt cardinal and pass upward through Be
rkeley cardinals (due to Woodin) and far beyond. B
agaria\, Woodin\, and myself have been charting ou
t this new hierarchy. I will discuss what we have
found so far.\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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