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The covering lemma up to a Woodin cardinal. (English) Zbl 0868.03021
\(K\) is the core model for one Woodin cardinal as introduced by J.R. Steel [The core model iterability problem (Lecture Notes Logic 8) (1996; Zbl 0864.03035)]. The authors prove the following covering lemma. \(\Omega\) is a measurable cardinal and there is no inner model with a Woodin cardinal. \(K\) is the core model constructed in \(V_\Omega\). \(\kappa<\Omega\) is a \(K\)-cardinal such that card(\(\kappa\)) is countably closed, i.e., for all \(\gamma<\text{card}(\kappa)\), \(\gamma^{\aleph_0}<\text{card}(\kappa)\). Let \(\lambda=(\kappa^+)^K\). Then \(\lambda<\kappa^+\) implies cf\((\lambda)=\text{card}(\kappa)\).
Thus in \(K\) successors of countably closed, singular cardinals are computed correctly. As part of the proof the authors investigate the fine structure of \(K\).

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