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13.
Ways of destruction
Farkas, Barnabás ; Zdomskyy, Lyubomyr, 1983-
We study the following natural strong variant of destroying Borel ideals: ▫$\mathbb{P}$▫ ▫$+\textit{-destroys}$▫ ▫$\mathcal{I}$▫ if ▫$\mathbb{P}$▫ adds an ▫$\mathcal{I}$▫-positive set which has ...finite intersection with every ▫$A\in\mathcal{I}\cap V$▫. Also, we discuss the associated variants ▫\begin{align*} \mathrm{non}^*(\mathcal{I},+)=&\min\big\{|\mathcal{Y}|:\mathcal{Y}\subseteq\mathcal{I}^+,\; \forall\;A\in\mathcal{I}\;\exists\;Y\in\mathcal{Y}\;|A\cap Y|<\omega\big\}\\ \mathrm{cov}^*(\mathcal{I},+)=&\min\big\{|\mathcal{C}|:\mathcal{C}\subseteq\mathcal{I},\; \forall\;Y\in\mathcal{I}^+\;\exists\;C\in\mathcal{C}\;|Y\cap C|=\omega\big\} \end{align*}▫ of the star-uniformity and the star-covering numbers of these ideals. Among other results, (1) we give a simple combinatorial characterisation when a real forcing ▫$\mathbb{P}_I$▫ can ▫$+$▫-destroy a Borel ideal ▫$\mathcal{J}$▫; (2) we discuss many classical examples of Borel ideals, their ▫$+$▫-destructibility, and cardinal invariants; (3) we show that the Mathias-Prikry, ▫$\mathbb{M}(\mathcal{I}^*)$▫-generic real ▫$+$▫-destroys ▫$\mathcal{I}$▫ iff ▫$\mathbb{M}(\mathcal{I}^*)$▫ ▫$+$▫-destroys ▫$\mathcal{I}$▫ iff ▫$\mathcal{I}$▫ can be ▫$+$▫-destroyed iff ▫$\mathrm{cov}^*(\mathcal{I},+)>\omega$▫; (4) we characterise when the Laver-Prikry, ▫$\mathbb{L}(\mathcal{I}^*)$▫-generic real ▫$+$▫-destroys ▫$\mathcal{I}$▫, and in the case of P-ideals, when exactly ▫$\mathbb{L}(\mathcal{I}^*)$▫ ▫$+$▫-destroys ▫$\mathcal{I}$▫; (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal.
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