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 $\lbrace A_\alpha :\alpha <\kappa \rbrace \subset \mathcal \{I\}$ 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 sets; meager sets; Baire space; additivity; 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 $\lbrace A_\alpha :\alpha <\kappa \rbrace \subset \mathcal {I}$ 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 sets; meager sets; Baire space; additivity; cardinal invariants; reals; pcf theory; null set; meager set; Baire space; additivity
UR - http://eudml.org/doc/246401
ER -