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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:
03E45 Inner models, including constructibility, ordinal definability, and core models
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