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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:
  Devlin, K.J., Constructibility, (1984), Springer Berlin · Zbl 0542.03029  Dodd, A.J., The core model, () · Zbl 0535.03029  Dodd, A.J.; Jensen, R.B., The core model, Ann. math. logic, 20, 43-75, (1981) · Zbl 0457.03051  Dodd, A.J.; Jensen, R.B., The covering lemma for K, Ann. math. logic, 22, 1-30, (1982) · Zbl 0492.03014  Dodd, A.J.; Jensen, R.B., The covering lemma for L[U], Ann. math. logic, 22, 127-135, (1982) · Zbl 0522.03037  Jensen, R.B., The fine structure of the constructible hierarchy, Ann. math. logic, 4, 229-308, (1972) · Zbl 0257.02035  R.B. Jensen, Non overlapping extenders, preprint.  Martin, D.A.; Steel, J.R., Iteration trees, J. amer. math. soc., 7, 1-73, (1994) · Zbl 0808.03035  Mitchell, W.J., The core model for sequences of measures I, (), 229-260 · Zbl 0539.03030  W.J. Mitchell, The core model for sequences of measures II, preprint. · Zbl 0539.03030  Mitchell, W.J.; Schimmerling, E., Covering without countable closure, Math. res. lett, 2, 595-609, (1995) · Zbl 0847.03024  Mitchell, W.J.; Steel, J.R., Fine structure and iteration trees, () · Zbl 0805.03042  Schimmerling, E., Combinatorial principles in the core model for one Woodin cardinal, Ann. pure appl. logic, 74, 153-201, (1995) · Zbl 0834.03018  E. Schimmerling, □_{κ}<ω holds in L[bdE], preprint.  E. Schimmerling and J.R. Steel, The Maximality of the core model, preprint. · Zbl 0928.03059  Schimmerling, E.; Steel, J.R., Fine structure for tame inner models, J. symbolic logic, 61, (1996) · Zbl 0861.03040  Steel, J.R., The wellfoundedness of the mitchell order on extenders, J. symbolic logic, 58, 931-940, (1993) · Zbl 0804.03038  Steel, J.R., Inner models with many Woodin cardinals, Ann. pure appl. logic, 65, 185-209, (1993) · Zbl 0805.03043  J.R. Steel, Core models with more Woodin cardinals, preprint. · Zbl 1012.03055  J.R. Steel, The core model iterability problem, Lecture Notes in Logic (Springer, Berlin) to appear. · Zbl 0864.03035  W.H. Woodin, Lectures at the 1995 Greater Boston Logic Conf, Workshop on Inner Model Theory.
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