More on cardinal invariants of analytic $P$-ideals

Farkas, Barnabás, and Soukup, Lajos. "More on cardinal invariants of analytic $P$-ideals." Commentationes Mathematicae Universitatis Carolinae 50.2 (2009): 281-295. <http://eudml.org/doc/32499>.

@article{Farkas2009,
abstract = {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 $\lbrace n\in \omega :f(n)> g(n)\rbrace \in \mathcal \{I\}$. 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 $\mathcal \{I\}$-bounding iff $\forall \, x\in \mathcal \{I\}\cap V^\{\mathbb \{P\}\}$$\exists \, y\in \mathcal \{I\}\cap V$$x\subseteq y$. $\mathbb \{P\}$ is $\mathcal \{I\}$-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.},
author = {Farkas, Barnabás, Soukup, Lajos},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {analytic $P$-ideals; cardinal invariants; forcing; analytic -ideal; cardinal invariant; forcing},
language = {eng},
number = {2},
pages = {281-295},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {More on cardinal invariants of analytic $P$-ideals},
url = {http://eudml.org/doc/32499},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Farkas, Barnabás
AU - Soukup, Lajos
TI - More on cardinal invariants of analytic $P$-ideals
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 2
SP - 281
EP - 295
AB - 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 $\lbrace n\in \omega :f(n)> g(n)\rbrace \in \mathcal {I}$. 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 $\mathcal {I}$-bounding iff $\forall \, x\in \mathcal {I}\cap V^{\mathbb {P}}$$\exists \, y\in \mathcal {I}\cap V$$x\subseteq y$. $\mathbb {P}$ is $\mathcal {I}$-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.
LA - eng
KW - analytic $P$-ideals; cardinal invariants; forcing; analytic -ideal; cardinal invariant; forcing
UR - http://eudml.org/doc/32499
ER -