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The self-iterability of L[E]

  • Ralf Schindler (a1) and John Steel (a2)

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 holds true, and if in addition λ is regular, then holds true.

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[1] D. Donder and P. Matet , Two cardinal versions of diamond, Israel Journal of Mathematics, vol. 83 (1993), pp. 143.

[5] W. Mitchell and E. Schimmerling , Weak covering without countable closure, Mathematical Research Letters, vol. 2 (1995), pp. 595609.

[6] W. Mitchell , E. Schimmerling , and J. Steel , The covering lemma up to a Woodin cardinal. Annals of Pure and Applied Logic, vol. 84 (1997), pp. 219255.

[9] S. Shelah , Around classification theory, Lecture Notes in Mathematics, no. 1182, Springer-Verlag, Berlin, 1986.

[10] J. Steel , The core model iterability problem, Lecture Notes in Logic, no. 8, Springer Verlag, 1996.

[15] M. Zeman , Inner models and large cardinals, Series in Logic and its Application, no. 5, de Gruyter, Berlin, New York, 2002.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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