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 > ω₁ are cardinals, then ◇_κ,λ^* holds true, and if in addition λ is regular, then ◇_κ,λ⁺ holds true.