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analytic $P$-ideals; cardinal invariants; forcing
Given an ideal $\mathcal I$ on $\omega $ let $\mathfrak{a} (\mathcal I)$ ($\bar{\mathfrak{a}}(\mathcal I)$) be minimum of the cardinalities of infinite (uncountable) maximal $\mathcal I$-almost disjoint subsets of $[{\omega}]^{\omega}$. We show that $\mathfrak{a} (\mathcal I_h)>{\omega}$ if $\mathcal I_h$ is a summable ideal; but $\mathfrak{a} ({\mathcal Z_{\vec \mu }})= {\omega}$ for any tall density ideal $\mathcal Z_{\vec \mu }$ including the density zero ideal $\mathcal Z$. On the other hand, you have $\mathfrak{b}\le \bar{\mathfrak{a}}(\mathcal I)$ for any analytic $P$-ideal $\mathcal I$, and $\bar{\mathfrak{a}}(\mathcal Z_{\vec \mu })\le \mathfrak{a}$ for each density ideal $\mathcal Z_{\vec \mu }$. For each ideal $\mathcal I$ on $\omega $ denote $\mathfrak{b}_{\mathcal I}$ and $\mathfrak{d}_{\mathcal I}$ the unbounding and dominating numbers of $\langle \omega ^\omega , \le_{\mathcal I}\rangle $ where $f\le_{\mathcal I} g$ iff $\{n\in \omega :f(n)> g(n)\}\in \mathcal I$. We show that $\mathfrak{b}_{\mathcal I}= \mathfrak{b}$ and $\mathfrak{d}_{\mathcal I}= \mathfrak{d}$ for each analytic $P$-ideal $\mathcal I$. Given a Borel ideal $\mathcal I$ on $\omega $ we say that a poset $\mathbb P$ is {\em $\mathcal I$-bounding\/} iff $\forall\, x\in \mathcal I\cap V^{\mathbb P}$ $\exists\, y\in \mathcal I\cap V$ $x\subseteq y$. $\mathbb P$ is {\em $\mathcal I$-dominating\/} iff $\exists\, y\in \mathcal I\cap V^{\mathbb P}$ $\forall\, x\in \mathcal I\cap V$ $x\subseteq^* y$. For each analytic $P$-ideal $\mathcal I$ if a poset $\mathbb P$ has the Sacks property then $\mathbb P$ is $\mathcal I$-bounding; moreover if $\mathcal I$ is tall as well then the property $\mathcal I$-bounding/$\mathcal I$-dominating implies ${\omega}^{\omega}$-bounding/adding dominating reals, and the converses of these two implications are false. For the density zero ideal $\mathcal Z$ we can prove more: (i) a poset $\mathbb P$ is $\mathcal Z$-bounding iff it has the Sacks property, (ii) if $\mathbb P$ adds a slalom capturing all ground model reals then $\mathbb P$ is $\mathcal Z$-dominating.
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