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A generalisation of closed unbounded sets

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Mathematical, Foundational and Computational Aspects of the Higher Infinite

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 unbounded (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 cardinals. Many of the simpler properties generalise completely, but for others we need an extra assumption. For instance to generalise the splitting property of stationary sets we have: $ begin{theorem}$ If $ kappa$ is $ Pi1$ $n$$-1$ indescribable, then any $n$-stationary subset of $ kappa$ is the union of $ kappa$ many pairwise-disjoint $n$-stationary sets. $ end{theorem}$ In $L$ these properties generalise straightforwardly as there any cardinal which admits an $n$-stationary set is $ Pi1_{n-1}$ indescribable (see $ cite{1}$).

If there is time I will also introduce a generalisation of ineffable cardinals and a weak $diamond$ principal that is associated.

[1] J. Bagaria, M. Magidor, and H. Sakai. Reflection and indescribability in the constructible universe. $ extit{Israel Journal of Mathematics}$, to appear (2012).

This talk is part of the Isaac Newton Institute Seminar Series series.

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