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A characterization of \({\square(\kappa^{+})}\) in extender models

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Abstract

We prove that, in any fine structural extender model with Jensen’s λ-indexing, there is a \({\square(\kappa^{+})}\) -sequence if and only if there is a pair of stationary subsets of \({\kappa^{+} \cap {\rm {cof}}( < \kappa)}\) without common reflection point of cofinality \({ < \kappa}\) which, in turn, is equivalent to the existence of a family of size \({ < \kappa}\) of stationary subsets of \({\kappa^{+} \cap {\rm {cof}}( < \kappa)}\) without common reflection point of cofinality \({ < \kappa}\) . By a result of Burke/Jensen, \({\square_\kappa}\) fails whenever \({\kappa}\) is a subcompact cardinal. Our result shows that in extender models, it is still possible to construct a canonical \({\square(\kappa^{+})}\) -sequence where \({\kappa}\) is the first subcompact.

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Correspondence to Martin Zeman.

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Research supported in part by NSF grant DMS-0500799. The authors would like to thank the anonymous referee for careful reading the paper and pointing out a confusing error.

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Kypriotakis, K., Zeman, M. A characterization of \({\square(\kappa^{+})}\) in extender models. Arch. Math. Logic 52, 67–90 (2013). https://doi.org/10.1007/s00153-012-0307-6

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  • DOI: https://doi.org/10.1007/s00153-012-0307-6

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