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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:
 3e+45 Inner models, including constructibility, ordinal definability, and core models 3e+55 Large cardinals
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##### References:
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