| 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:
|
http://hdl.handle.net/10338.dmlcz/141435 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
