The Journal of Symbolic Logic

Research Article

The self-iterability of L[E]

Ralf Schindlera1 and John Steela2

a1 Institut für Mathematische Logik und Grundlagenforschung, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany, E-mail:

a2 Department of Mathematics, 717 Evans Hall, University of California, Berkeley, Ca 94720, USA, E-mail:


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.

(Received February 20 2007)