The self-iterability of L [ E ]

Abstract

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 κ which is not a limit of Woodin cardinals there is some cutpoint t < κ such that J κ [ E ] is iterable above t with respect to iteration trees of length less than κ . As an application we show L [ E ] to be a model of the following two cardinals versions of the diamond principle. If λ > κ > ω 1 are cardinals, then holds true, and if in addition λ is regular, then holds true.