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Square in core models. (English) Zbl 0992.03062
Summary: We prove that in all Mitchell-Steel core models, \(\square_\kappa\) holds for all \(\kappa\). From this we obtain new consistency strength lower bounds for the failure of \(\square_\kappa\) if \(\kappa\) is either singular and countably closed, weakly compact, or measurable. Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, \(\square_\kappa\) holds if \(\kappa\) is not subcompact. (See Theorem 15; the only-if direction is essentially due to Jensen).

MSC:
03E45 Inner models, including constructibility, ordinal definability, and core models
03E55 Large cardinals
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