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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 \(K\).

03E55 Large cardinals
03E45 Inner models, including constructibility, ordinal definability, and core models
Full Text: DOI
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