A Sacks amoeba preserving distributivity of $\mathcal {P}(\omega )/\mathrm {fin}$
Volume 254 / 2021
Fundamenta Mathematicae 254 (2021), 261-303
MSC: Primary 03E17; Secondary 03E35.
DOI: 10.4064/fm961-9-2020
Published online: 16 February 2021
Abstract
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.
