Square in core models

Bulletin of Symbolic Logic 7 (3):305-314 (2001)
We prove that in all Mitchell-Steel core models, □ κ holds for all κ. (See Theorem 2.). From this we obtain new consistency strength lower bounds for the failure of □ κ if κ is either singular and countably closed, weakly compact, or measurable. (Corallaries 5, 8, and 9.) 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, □ κ holds iff κ is not subcompact. (See Theorem 15; the only if direction is essentially due to Jensen.)
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DOI 10.2307/2687750
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Jason Aaron Schanker (2013). Partial Near Supercompactness. Annals of Pure and Applied Logic 164 (2):67-85.
Martin Zeman (2010). Global Square Sequences in Extender Models. Annals of Pure and Applied Logic 161 (7):956-985.

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