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## Controlling classical cardinal characteristics while collapsing cardinals

Colloquium Mathematicum MSC: Primary 03E17; Secondary 03E35, 03E40. DOI: 10.4064/cm8420-2-2022 Opublikowany online: 20 May 2022

#### Streszczenie

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.

#### Autorzy

• 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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