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SUMMARY:The Search for Deep Inconsistency - Koellner\, P (Harvard Universi
 ty)
DTSTART:20150825T080000Z
DTEND:20150825T090000Z
UID:TALK60450@talks.cam.ac.uk
CONTACT:42080
DESCRIPTION:The hierarchy of large cardinals provides us with a canonical 
 means to climb the hierarchy of consistency strength. There have been  any
  purported inconsistency proofs of various large cardinal axioms. For exam
 ple\, there have been many proofs purporting to show that measurable cardi
 nals are inconsistent. But to date the only proofs that have stood the tes
 t of time are those which are rather transparent and simple\, the most not
 able example being Kunen's proof showing that Reinhardt cardinals are inco
 nsistent. The Kunen result\, however\, makes use of AC. And long standing 
 open question is whether Reinhardt cardinals are consistent in the context
  of ZF.\n\nIn this talk I will survey the simple inconsistency proofs and 
 then raise the question of whether perhaps the large cardinal hierarchy ou
 tstrips AC\, passing through Reinhardt cardinals and reaching far beyond. 
 There are two main motivations for this investigation. First\, it is of in
 terest in its own right to determine whether the hierarchy of consistency 
 strength outstrips AC. Perhaps there is an entire "choicless" large cardin
 al hierarchy\, one which reaches new consistency strengths and has fruitfu
 l applications. Second\, since the task of proving an inconsistency result
  becomes easier as one strengthens the hypothesis\, in the search for a de
 ep inconsistency it is reasonable to start with outlandishly strong large 
 cardinal assumptions and then work ones way down. This will lead to the fo
 rmulation of large cardinal axioms (in the context of ZF) that start at th
 e level of a Reinhardt cardinal and pass upward through Berkeley cardinals
  (due to Woodin) and far beyond. Bagaria\, Woodin\, and myself have been c
 harting out this new hierarchy. I will discuss what we have found so far.\
 n
LOCATION:Seminar Room 1\, Newton Institute
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