Journal of Symbolic Logic

The self-iterability of L[E]

Ralf Schindler and John Steel

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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 Jκ[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.

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J. Symbolic Logic, Volume 74, Issue 3 (2009), 751-779.

First available in Project Euclid: 16 June 2009

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Schindler, Ralf; Steel, John. The self-iterability of L[E]. J. Symbolic Logic 74 (2009), no. 3, 751--779. doi:10.2178/jsl/1245158084.

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