Uniformity numbers of the null-additive and meager-additive ideals
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| Format: | Preprint |
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2024
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| author | Cardona, Miguel A. Mejía, Diego A. Rivera-Madrid, Ismael E. |
| author_facet | Cardona, Miguel A. Mejía, Diego A. Rivera-Madrid, Ismael E. |
| contents | Denote by $\mathcal{NA}$ and $\mathcal{MA}$ the ideals of null-additive and meager-additive subsets of~$2^ω$, 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 $\mathfrak{b}< \mathrm{add}(\mathcal{SN})<\mathrm{cov}(\mathcal{M})<\mathfrak{d}=\mathfrak{c}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_15364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniformity numbers of the null-additive and meager-additive ideals Cardona, Miguel A. Mejía, Diego A. Rivera-Madrid, Ismael E. Logic 03E05, 03E17, 03E35, 03E40 Denote by $\mathcal{NA}$ and $\mathcal{MA}$ the ideals of null-additive and meager-additive subsets of~$2^ω$, 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 $\mathfrak{b}< \mathrm{add}(\mathcal{SN})<\mathrm{cov}(\mathcal{M})<\mathfrak{d}=\mathfrak{c}$. |
| title | Uniformity numbers of the null-additive and meager-additive ideals |
| topic | Logic 03E05, 03E17, 03E35, 03E40 |
| url | https://arxiv.org/abs/2401.15364 |