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Title: Pcf theory and cardinal invariants of the reals (English)
Author: Soukup, Lajos
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 52
Issue: 1
Year: 2011
Pages: 153-162
Summary lang: English
Category: math
Summary: 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 $A=\operatorname{pcf}(A)$. (English)
Keyword: cardinal invariants
Keyword: reals
Keyword: pcf theory
Keyword: null sets
Keyword: meager sets
Keyword: Baire space
Keyword: additivity
MSC: 03E04
MSC: 03E17
MSC: 03E35
idZBL: Zbl 1240.03021
idMR: MR2828366
Date available: 2011-03-08T17:45:04Z
Last updated: 2013-09-22
Stable URL:
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