Controlling classical cardinal characteristics while collapsing cardinals

Martin Goldstern, Jakob Kellner, Diego A. Mejía, Saharon Shelah Colloquium Mathematicum MSC: Primary 03E17; Secondary 03E35, 03E40. DOI: 10.4064/cm8420-2-2022 Published online: 20 May 2022

Abstract

We show how to force distinct values to $\mathfrak m$, $\mathfrak p$ and $\mathfrak h$ and the values in Cichoń’s diagram, using the Boolean Ultrapower method. In our recent paper [J. Math. Logic 21 (2021)] the same was done for a newer Cichoń’s Maximum construction which does not require large cardinals. The present version does need large cardinals, but allows one more value, in addition to the continuum, to be singular (either $\mathrm {cov}(\mathcal M)$ or $\mathfrak d$).

We also show the following: Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we can compose $P$ with a collapse (of a cardinal $\lambda \gt \kappa $ to $\kappa $) such that the composition still forces the previous values to these characteristics.

Authors

  • Martin GoldsternInstitut für Diskrete Mathematik
    und Geometrie
    Technische Universität Wien
    Wiedner Hauptstrasse 8-10/104
    1040 Wien, Austria
    e-mail
  • Jakob KellnerInstitut für Diskrete Mathematik und Geometrie
    Technische Universität Wien
    Wiedner Hauptstrasse 8-10/104
    1040 Wien, Austria
    e-mail
  • Diego A. MejíaFaculty of Science
    Creative Science Course (Mathematics)
    Shizuoka University
    Ohya 836, Suruga-ku
    Shizuoka-shi, Japan 422-8529
    e-mail
  • Saharon ShelahEinstein Institute of Mathematics
    Edmond J. Safra Campus, Givat Ram
    The Hebrew University of Jerusalem
    Jerusalem, 91904, Israel
    and
    Department of Mathematics
    Rutgers University
    New Brunswick, NJ 08854, USA
    e-mail

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