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A Sacks amoeba preserving distributivity of $\mathcal {P}(\omega )/\mathrm {fin}$

Volume 254 / 2021

Otmar Spinas, Wolfgang Wohofsky Fundamenta Mathematicae 254 (2021), 261-303 MSC: Primary 03E17; Secondary 03E35. DOI: 10.4064/fm961-9-2020 Published online: 16 February 2021


By iterating an increasing amoeba for Sacks forcing (implicitly introduced by Louveau, Shelah, and Veličković), we obtain a model in which $\mathfrak{h} $ (i.e., the distributivity of ${\mathcal {P}}(\omega )/\mathrm {fin}$) is smaller than the additivity of the Marczewski ideal (the ideal associated with Sacks forcing). The forcing is different from the usual amoeba for Sacks forcing: Unlike the latter, it has the pure decision and the Laver property, and therefore does not add Cohen reals. In our model, $\mathfrak{h} \lt \mathfrak{h} _\omega $ holds true, which answers a question by Repický who asked whether $\mathfrak{h} _\omega $ equals $\mathfrak{h} $ in ZFC.


  • Otmar SpinasMathematisches Seminar
    Christian-Albrechts-Universität zu Kiel
    Ludewig-Meyn-Straße 4
    24118 Kiel, Germany
  • Wolfgang WohofskyKurt Gödel Research Center
    for Mathematical Logic
    University of Vienna
    Augasse 2-6
    1090 Wien, Austria

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