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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
- ISSN
- 0933-5846
- eISSN
- 1432-0665
- DOI
- 10.1007/s00153-021-00808-0
- Publisher site
- See Article on Publisher Site
Abstract
We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal SN\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathcal {S}}}{{\mathcal {N}}}$$\end{document}. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that non(SN)<cov(SN)<cof(SN)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathrm {non}({{\mathcal {S}}}{{\mathcal {N}}})<\mathrm {cov}({{\mathcal {S}}}{{\mathcal {N}}})<\mathrm {cof}({{\mathcal {S}}}{{\mathcal {N}}})$$\end{document}, which is the first consistency result where more than two cardinal invariants associated with SN\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathcal {S}}}{{\mathcal {N}}}$$\end{document} are pairwise different. Another consequence is that SN⊆s0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathcal {S}}}{{\mathcal {N}}}\subseteq s^0$$\end{document} in ZFC where s0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s^0$$\end{document} denotes Marczewski’s ideal.
Journal
Archive for Mathematical Logic – Springer Journals
Published: Jul 1, 2022
Keywords: Strong measure zero sets; Cardinal invariants; Sacks model; Yorioka ideals; 03E17; 03E35; 03E40