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SUMMARY:A generalisation of closed unbounded sets - Brickhill\, H (Univers
 ity of Bristol)
DTSTART:20150824T123000Z
DTEND:20150824T130000Z
UID:TALK60433@talks.cam.ac.uk
CONTACT:42080
DESCRIPTION:A generalisation of stationarity\, associated with stationary 
 reflection\, was introduced in [1]. I give an alternative characterisation
  of these $n$-stationary sets by defining a generalisation of closed unbou
 nded (club) sets\, so an $n$-stationary set is defined in terms of these $
 n$-clubs in the usual way. I will then look into what familiar properties 
 of stationary and club sets will still hold in this more general setting\,
  and explore the connection between these concepts and indescribable cardi
 nals. Many of the simpler properties generalise completely\, but for other
 s we need an extra assumption. For instance to generalise the splitting pr
 operty of stationary sets we have: $ begin{theorem}$ If $ kappa$ is $ Pi^1
 $ $n$$-1$ indescribable\, then any $n$-stationary subset of $ kappa$ is th
 e union of $ kappa$ many pairwise-disjoint $n$-stationary sets. $ end{theo
 rem}$ In $L$ these properties generalise straightforwardly as there any ca
 rdinal which admits an $n$-stationary set is $ Pi^1_{n-1}$ indescribable (
 see $ cite{1}$).\n\nIf there is time I will also introduce a generalisatio
 n of ineffable cardinals and a weak $diamond$ principal that is associated
 .\n\n[1] J. Bagaria\, M. Magidor\, and H. Sakai. Reflection and indescriba
 bility in the constructible universe. $	extit{Israel Journal of Mathematic
 s}$\, to appear (2012).\n\n
LOCATION:Seminar Room 1\, Newton Institute
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