Inner models with many Woodin cardinals.

*(English)*Zbl 0805.03043This paper extends the author’s work on fine structure and iteration trees [see the review above] to models with more than one Woodin cardinal.

The main result of the paper is: Assume there are (in order type) \(\theta\)-many Woodin cardinals. Then there is a good extender sequence \(\vec E\) such that (1) \(L[\vec E]\models\) “there are \(\theta\) Woodin cardinals”, (2) every level \(J^{\vec E}_ \alpha\) of \(L[\vec E]\) is an \(\omega\)-sound, meek premouse, (3) \(L[\vec E]\models\text{GCH}\).

The paper concludes with a discussion of minimal models which also satisfy “there are \(\omega\) Woodin cardinals”. The paper announces results to appear elsewhere of the case where there are \(n\) Woodin cardinals.

The main result of the paper is: Assume there are (in order type) \(\theta\)-many Woodin cardinals. Then there is a good extender sequence \(\vec E\) such that (1) \(L[\vec E]\models\) “there are \(\theta\) Woodin cardinals”, (2) every level \(J^{\vec E}_ \alpha\) of \(L[\vec E]\) is an \(\omega\)-sound, meek premouse, (3) \(L[\vec E]\models\text{GCH}\).

The paper concludes with a discussion of minimal models which also satisfy “there are \(\omega\) Woodin cardinals”. The paper announces results to appear elsewhere of the case where there are \(n\) Woodin cardinals.

Reviewer: A.M.Coyne (Asheville)

##### MSC:

03E55 | Large cardinals |

##### Keywords:

fine structure; iteration trees; models with more than one Woodin cardinal; extender sequence; minimal models
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\textit{J. R. Steel}, Ann. Pure Appl. Logic 65, No. 2, 185--209 (1993; Zbl 0805.03043)

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##### References:

[1] | D.A. Martin and J.R. Steel, Iteration trees, in the J. AMS, to appear · Zbl 0808.03035 |

[2] | J.R. Steel, Projectively wellordered inner models, to appear |

[3] | W.J. Mitchell, Embeddings of iteation trees, unpublished notes |

[4] | W.J. Mitchell and J.R. Steel, Fine structure and iteration trees, ASL Lecture Notes in Logic, to appear |

[5] | Schimmerling, E., Combinatorial principles in the core model, Ph.D. thesis, (1992), UCLA |

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