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On Semi-Proper Forcing

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Part of the Lecture Notes in Mathematics book series (LNM,volume 940)

Abstract

We weaken the notion of proper to semi-proper, so that the important properties (e.g., being preserved by some iterations) are preserved and it includes some forcing which changes the cofinality of a regular cardinal > 1 to 0. So, using the right iterations, we can iterate such forcing without collapsing 1. As a result, we solve the following problems of Friedman, Magidor and Avraham, by proving (modulo large cardinals) the consistency of the following with G.C.H.:

  1. (1)

    for every S 2, S or 2S contains a closed copy of ω 1,

  2. (2)

    there is a normal precipitous filter D on 2, {δ < 2: cf δ = 0} ∈ D,

  3. (3)

    for every A ⊆ N2, {δ < 2: cf δ = 0, δ is regular in L(δA)} is stationary.

Keywords

  • Chain Condition
  • Winning Strategy
  • Regular Cardinal
  • Measurable Cardinal
  • Countable Support

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1982 Springer-Verlag Berlin Heidelberg

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Shelah, S. (1982). On Semi-Proper Forcing. In: Proper Forcing. Lecture Notes in Mathematics, vol 940. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21543-2_10

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  • DOI: https://doi.org/10.1007/978-3-662-21543-2_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11593-9

  • Online ISBN: 978-3-662-21543-2

  • eBook Packages: Springer Book Archive