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