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The self-iterability of $$L[E]$$. (English) Zbl 1178.03067
Summary: Let $$L[E]$$ be an iterable tame extender model. We analyze to which extent $$L[E]$$ knows fragments of its own iteration strategy. Specifically, we prove that, inside $$L[E]$$, for every cardinal $$\kappa$$ which is not a limit of Woodin cardinals there is some cutpoint $$t<\kappa$$ such that $$J_\kappa[E]$$ is iterable above $$t$$ with respect to iteration trees of length less than $$\kappa$$.
As an application we show $$L[E]$$ to be a model of the following two-cardinals versions of the diamond principle. If $$\lambda>\kappa>\omega_1$$ are cardinals, then $$\lozenge_{\kappa,\lambda}^*$$ holds true, and if in addition $$\lambda$$ is regular, then $$\lozenge_{\kappa,\lambda}^+$$ holds true.

##### MSC:
 3e+45 Inner models, including constructibility, ordinal definability, and core models
##### Keywords:
iterable tame extender model; cutpoint; iteration trees; diamond
Full Text:
##### References:
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