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UNIFORMITY NUMBERS OF THE NULL-ADDITIVE AND MEAGER-ADDITIVE IDEALS
Part of:
Set theory
Published online by Cambridge University Press: 07 August 2025
Abstract
Denote by 
$\mathcal {NA}$ and 
$\mathcal {MA}$ the ideals of null-additive and meager-additive subsets of 
$2^{\omega }$, respectively. We prove in ZFC that 
$\mathrm {add}(\mathcal {NA})=\mathrm {non}(\mathcal {NA})$ and introduce a new (Polish) relational system to reformulate Bartoszyński’s and Judah’s characterization of the uniformity of 
$\mathcal {MA}$, which is helpful to understand the combinatorics of 
$\mathcal {MA}$ and to prove consistency results. As for the latter, we prove that 
$\mathrm {cov}(\mathcal {MA})<\mathfrak {c}$ (even 
$\mathrm {cov}(\mathcal {MA})<\mathrm {non}(\mathcal {N})$) is consistent with ZFC, as well as several constellations of Cichoń’s diagram with 
$\mathrm {non}(\mathcal {NA})$, 
$\mathrm {non}(\mathcal {MA}),$ and 
$\mathrm {add}(\mathcal {SN})$, which include 
$\mathrm {non}(\mathcal {NA})<\mathfrak {b}< \mathrm {non}(\mathcal {MA})$ and 
$\mathfrak {b}< \mathrm {add}(\mathcal {SN})<\mathrm {cov}(\mathcal {M})<\mathfrak {d}=\mathfrak {c}$.
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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