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Iterated forcing and normal ideals onω 1

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Abstract

We prove that suitable iteration does not collapse ℵ1 [and does not add reals], i.e., that in such iteration, certain sealing of maximal antichains of stationary subsets ofω 1 is allowed. As an application, e.g., we prove from supercompact hypotheses, mainly, the consistency of: ZFC + GCH + “for some stationary setS ⊆ω 1, {ie345-1}(ω 1)/(D ω 1 +S) is the Levy algebra” (i.e., the complete Boolean Algebra corresponding to the Levy collapse Levy (ℵ0,<ℵ2) (and we can add “a variant of PFA”) and the consistency of the same, with “Ulam property” replacing “Levy algebra”). The paper assumes no specialized knowledge (if you agree to believe in the semi-properness iteration theorem and RCS iteration).

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References

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    Saharon Shelah

  2. Department of Mathematics and EECS, University of Michigan, Ann Arbor, Michigan, USA

    Saharon Shelah

  3. Department of Mathematics, Rutgers University, New Brunswick, New Jersey, USA

    Saharon Shelah

Authors
  1. Saharon Shelah
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Additional information

This research was partially supported by the NSF.

This paper was largely written during the author’s visit at Cal Tech around the end of April 1985. The author would like to thank M. Foreman, A. Kekris and H. Woodin for their hospitality.

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Shelah, S. Iterated forcing and normal ideals onω 1 . Israel J. Math. 60, 345–380 (1987). https://doi.org/10.1007/BF02780398

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  • DOI: https://doi.org/10.1007/BF02780398

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