Pcf theory and cardinal invariants of the reals

Soukup, Lajos. "Pcf theory and cardinal invariants of the reals." Commentationes Mathematicae Universitatis Carolinae 52.1 (2011): 153-162. <http://eudml.org/doc/246401>.

@article{Soukup2011,
abstract = {The additivity spectrum $\operatorname\{ADD\}(\mathcal\{I\})$ of an ideal $\mathcal\{I\}\subset \mathcal\{P\}(I)$ is the set of all regular cardinals $\kappa$ such that there is an increasing chain $\\{A_\alpha:\alpha<\kappa\\}\subset \mathcal\{I\}$ with $\bigcup_\{\alpha<\kappa\}A_\alpha\notin \mathcal\{I\}$. We investigate which set $A$ of regular cardinals can be the additivity spectrum of certain ideals. Assume that $\mathcal\{I\}=\mathcal\{B\}$ or $\mathcal\{I\}=\mathcal\{N\}$, where $\mathcal\{B\}$ denotes the $\{\sigma\}$-ideal generated by the compact subsets of the Baire space $\omega^\omega$, and $\mathcal\{N\}$ is the ideal of the null sets. We show that if $A$ is a non-empty progressive set of uncountable regular cardinals and $\operatorname\{pcf\}(A)=A$, then $\operatorname\{ADD\}(\mathcal\{I\})=A$ in some c.c.c generic extension of the ground model. On the other hand, we also show that if $A$ is a countable subset of $\operatorname\{ADD\}(\mathcal\{I\})$, then $\operatorname\{pcf\}(A)\subset \operatorname\{ADD\}(\mathcal\{I\})$. For countable sets these results give a full characterization of the additivity spectrum of $\mathcal\{I\}$: a non-empty countable set $A$ of uncountable regular cardinals can be $\operatorname\{ADD\}(\mathcal\{I\})$ in some c.c.c generic extension iff $A=\operatorname\{pcf\}(A)$.},
author = {Soukup, Lajos},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {cardinal invariants; reals; pcf theory; null set; meager set; Baire space; additivity},
language = {eng},
number = {1},
pages = {153-162},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Pcf theory and cardinal invariants of the reals},
url = {http://eudml.org/doc/246401},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Soukup, Lajos
TI - Pcf theory and cardinal invariants of the reals
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 1
SP - 153
EP - 162
AB - The additivity spectrum $\operatorname{ADD}(\mathcal{I})$ of an ideal $\mathcal{I}\subset \mathcal{P}(I)$ is the set of all regular cardinals $\kappa$ such that there is an increasing chain $\{A_\alpha:\alpha<\kappa\}\subset \mathcal{I}$ with $\bigcup_{\alpha<\kappa}A_\alpha\notin \mathcal{I}$. We investigate which set $A$ of regular cardinals can be the additivity spectrum of certain ideals. Assume that $\mathcal{I}=\mathcal{B}$ or $\mathcal{I}=\mathcal{N}$, where $\mathcal{B}$ denotes the ${\sigma}$-ideal generated by the compact subsets of the Baire space $\omega^\omega$, and $\mathcal{N}$ is the ideal of the null sets. We show that if $A$ is a non-empty progressive set of uncountable regular cardinals and $\operatorname{pcf}(A)=A$, then $\operatorname{ADD}(\mathcal{I})=A$ in some c.c.c generic extension of the ground model. On the other hand, we also show that if $A$ is a countable subset of $\operatorname{ADD}(\mathcal{I})$, then $\operatorname{pcf}(A)\subset \operatorname{ADD}(\mathcal{I})$. For countable sets these results give a full characterization of the additivity spectrum of $\mathcal{I}$: a non-empty countable set $A$ of uncountable regular cardinals can be $\operatorname{ADD}(\mathcal{I})$ in some c.c.c generic extension iff $A=\operatorname{pcf}(A)$.
LA - eng
KW - cardinal invariants; reals; pcf theory; null set; meager set; Baire space; additivity
UR - http://eudml.org/doc/246401
ER -