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Inner models with many Woodin cardinals. (English) Zbl 0805.03043
This 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.

##### MSC:
 3e+55 Large cardinals
Full Text:
##### 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[2] J.R. Steel, Projectively wellordered inner models, to appear [3] W.J. Mitchell, Embeddings of iteation trees, unpublished notes [4] W.J. Mitchell and J.R. Steel, Fine structure and iteration trees, ASL Lecture Notes in Logic, to appear [5] Schimmerling, E., Combinatorial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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