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# Fine structure for tame inner models

## Extract

In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded models all of whose levels are tame (defined in [St2] and below). As a consequence, more powerful large cardinal properties reflect to fine structural inner models. For example, we get the following extension to [MiSt, Theorem 11.3] and [St2, Theorem 0.3].

Suppose that there is a strong cardinal that is a limit of Woodin cardinals. Then there is a good extender sequence such that

(1) every level of is a sound, tame mouse, and

(2) ⊨ “There is a strong cardinal that is a limit of Woodin cardinals”.

Recall that satisfies GCH if all its levels are sound. Another consequence of our work is the following covering property, an extension to [St1, Theorem 1.4] and [St3, Theorem 1.10].

Suppose that fi is a normal measure on Ω and that all premice are tame. Then Kc, the background certified core model, exists and is a premouse of height Ω. Moreover, for μ-almost every α < Ω.

Ideas similar to those introduced here allow us to extend the fine structure theory of [Sch] to the level of tame mice. The details of this extension shall appear elsewhere. From the extension of [Sch] and Theorem 0.2, new relative consistency results follow. For example, we have the following application.

If there is a cardinal κ such that κ is κ+-strongly compact, then there is a premouse that is not tame.

## References

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[MaSt]Martin, D. A. and Steel, J. R., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), no. 1, pp. 173.
[MiSt]Mitchell, W. J. and Steel, J. R., Fine structure and iteration trees, Springer Lecture Notes in Logic, vol. 3, Springer-Verlag, 1994.
[Sch]Schimmerling, E., Combinatorial principles in the core model for one Woodin cardinal, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 153201.
[Stl]Steel, J. R., The core model iterability problem, to appear in Springer Lecture Notes in Logic.
[St3]Steel, J. R., Core models with more Woodin cardinals, preprint.
[St2]Steel, J. R., Inner models with many Woodin cardinals, Annals of Pure and Applied Logic, vol. 65 (1993), pp. 185209.

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