Square in core models

Bulletin of Symbolic Logic 7 (3):305-314 (2001)
Abstract
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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References found in this work BETA
The Fine Structure of the Constructible Hierarchy.R. Björn Jensen - 1972 - Annals of Mathematical Logic 4 (3):229-308.
Combinatorial Principles in the Core Model for One Woodin Cardinal.Ernest Schimmerling - 1995 - Annals of Pure and Applied Logic 74 (2):153-201.
Smooth Categories and Global □.Ronald B. Jensen & Martin Zeman - 2000 - Annals of Pure and Applied Logic 102 (1-2):101-138.
A Finite Family Weak Square Principle.Ernest Schimmerling - 1999 - Journal of Symbolic Logic 64 (3):1087-1110.

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Citations of this work BETA
Inner Models in the Region of a Woodin Limit of Woodin Cardinals.Itay Neeman - 2002 - Annals of Pure and Applied Logic 116 (1-3):67-155.
Hierarchies of Forcing Axioms I.Itay Neeman & Ernest Schimmerling - 2008 - Journal of Symbolic Logic 73 (1):343-362.
Hierarchies of Forcing Axioms II.Itay Neeman - 2008 - Journal of Symbolic Logic 73 (2):522 - 542.
Weak Compactness and No Partial Squares.John Krueger - 2011 - Journal of Symbolic Logic 76 (3):1035 - 1060.

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