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WAYS OF DESTRUCTION

Part of: Set theory

Published online by Cambridge University Press:  08 October 2021

BARNABÁS FARKAS  and
LYUBOMYR ZDOMSKYY
BARNABÁS FARKAS
Affiliation:
IINSTITUT FÜR DISKRETE MATHEMATIK UND GEOMETRIE TECHNISCHE UNIVERSITÄT WIEN WIEDNER HAUPTSTRASSE 8-10/104 1040WIEN, AUSTRIAE-mail:barnabasfarkas@gmail.comURL: http://dmg.tuwien.ac.at/farkas/E-mail:lzdomsky@gmail.comURL: http://dmg.tuwien.ac.at/zdomskyy/
LYUBOMYR ZDOMSKYY
Affiliation:
IINSTITUT FÜR DISKRETE MATHEMATIK UND GEOMETRIE TECHNISCHE UNIVERSITÄT WIEN WIEDNER HAUPTSTRASSE 8-10/104 1040WIEN, AUSTRIAE-mail:barnabasfarkas@gmail.comURL: http://dmg.tuwien.ac.at/farkas/E-mail:lzdomsky@gmail.comURL: http://dmg.tuwien.ac.at/zdomskyy/
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Abstract

We study the following natural strong variant of destroying Borel ideals: $\mathbb {P}$ $+$ -destroys $\mathcal {I}$ if $\mathbb {P}$ adds an $\mathcal {I}$ -positive set which has finite intersection with every $A\in \mathcal {I}\cap V$ . Also, we discuss the associated variants

$$ \begin{align*} \mathrm{non}^*(\mathcal{I},+)=&\min\big\{|\mathcal{Y}|:\mathcal{Y}\subseteq\mathcal{I}^+,\; \forall\;A\in\mathcal{I}\;\exists\;Y\in\mathcal{Y}\;|A\cap Y|<\omega\big\},\\ \mathrm{cov}^*(\mathcal{I},+)=&\min\big\{|\mathcal{C}|:\mathcal{C}\subseteq\mathcal{I},\; \forall\;Y\in\mathcal{I}^+\;\exists\;C\in\mathcal{C}\;|Y\cap C|=\omega\big\} \end{align*} $$
of the star-uniformity and the star-covering numbers of these ideals.

Among other results, (1) we give a simple combinatorial characterisation when a real forcing $\mathbb {P}_I$ can $+$ -destroy a Borel ideal $\mathcal {J}$ ; (2) we discuss many classical examples of Borel ideals, their $+$ -destructibility, and cardinal invariants; (3) we show that the Mathias–Prikry, $\mathbb {M}(\mathcal {I}^*)$ -generic real $+$ -destroys $\mathcal {I}$ iff $\mathbb {M}(\mathcal {I}^*)\ +$ -destroys $\mathcal {I}$ iff $\mathcal {I}$ can be $+$ -destroyed iff $\mathrm {cov}^*(\mathcal {I},+)>\omega $ ; (4) we characterise when the Laver–Prikry, $\mathbb {L}(\mathcal {I}^*)$ -generic real $+$ -destroys $\mathcal {I}$ , and in the case of P-ideals, when exactly $\mathbb {L}(\mathcal {I}^*)$ $+$ -destroys $\mathcal {I}$ ; and (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal.

Keywords

Borel idealMathias–Prikry forcingLaver–Prikry forcingLaflamme’s forcingforcing indestructibilitycovering propertyTukey connectionKatětov orderfragile idealcardinal invariantsBorel Determinacytrace ideal

MSC classification

Primary: 03E05: Other combinatorial set theory
Secondary: 03E15: Descriptive set theory 03E17: Cardinal characteristics of the continuum 03E35: Consistency and independence results
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 3 , September 2022 , pp. 938 - 966
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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