A characterization of {\square(\kappa^{+})} in extender models |
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Authors: | Kyriakos Kypriotakis Martin Zeman |
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Institution: | 1. Southwestern Oregon Community College, 1988 Newmark Ave, Coos Bay, OR, 97420, USA 2. Department of Mathematics, University of California at Irvine, Irvine, CA, 92697, USA
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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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