Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-02-11T20:37:08.312Z Has data issue: false hasContentIssue false
  • English
  • Français

THE COVERING NUMBERS OF SOME MYCIELSKI IDEALS MAY BE DIFFERENT

Part of: Set theory

Published online by Cambridge University Press:  30 January 2025

OTMAR SPINAS
OTMAR SPINAS*
Affiliation:
MATHEMATISCHES SEMINAR CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL HEINRICH-HECHT-PLATZ 6 KIEL 24118, GERMANY
*
Email: spinas@math.uni-kiel.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that in the Silver model the inequality $\mathrm {cov}(\mathfrak {C} _2) < \mathrm {cov}(\mathfrak {P}_2)$ holds true, where $\mathfrak {C}_2$ and $\mathfrak {P}_2$ are the two-dimensional Mycielski ideals.

Keywords

Mycielski idealstree idealscovering coefficientsSilver forcingSacks treeuniform tree

MSC classification

Primary: 03E17: Cardinal characteristics of the continuum 03E35: Consistency and independence results
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 26
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baumgartner, J. E., Iterated Forcing: Surveys in Set Theory (Mathias, A. R. D., editor), London Mathematical Society Lecture Notes Series, 87, Cambridge University Press, London, 1983, pp. 1–59.Google Scholar
Cichoń, J., Rosłanowski, A., Steprāns, J., and Weglorz, B., Combinatorial properties of the ideal ${\mathfrak{P}}_2$ . Journal of Symbolic Logic , vol. 58 (1993), pp. 42–54.Google Scholar
Grigorieff, S., Combinatorics of ideals and forcing . Annals of Mathematical Logic , vol. 3 (1971), pp. 363–394.Google Scholar
Judah, H., Miller, A. W., and Shelah, S., Sacks forcing, Laver forcing, and Martin’s axiom . Archive for Mathematical Logic , vol. 31 (1992), pp. 145–161.Google Scholar
Kuiper, J. M. and Spinas, O., Mycielski ideals and uniform trees . Fundamenta Mathematicae , vol. 264 (2024), pp. 103–147.Google Scholar
Kuiper, J. M. and Spinas, O., Different covering numbers of compact tree ideals . Archive for Mathematical Logic , (2024). https://doi.org/10.1007/s00153-024-00933-6.Google Scholar
Mycielski, J., Some new ideals of sets on the real line . Colloquium Mathematicum , vol. 20 (1969), pp. 71–76.Google Scholar
Rosłanowski, A., On game ideals . Colloquium Mathematicum , vol. 59 (1990), no. 2, pp. 159–168.Google Scholar
Rosłanowski, A. and Steprāns, J., Chasing silver . Canadian Mathematical Bulletin , vol. 51 (2008), no. 4, pp. 593–603.Google Scholar
Shelah, S. and Steprāns, J., The covering numbers of Mycielski ideals are all equal . Journal of Symbolic Logic , vol. 66 (2001), no. 2, pp. 707–718.Google Scholar
Shelah, S., Proper and Improper Forcing , Springer, Berlin, 1997.Google Scholar