Almost disjoint refinements and mixing reals

Barnabás Farkas ¹   ; Yurii Khomskii ²   ; Zoltán Vidnyánszky ³

¹ Kurt Gödel Research Center for Mathematical Logic Vienna, Austria
² University of Hamburg Hamburg, Germany
³ Alfréd Rényi Institute of Mathematics Budapest, Hungary

Fundamenta Mathematicae, Tome 242 (2018) no. 1, pp. 25-48

Résumé

We investigate families of subsets of $\omega $ with almost disjoint refinements in the classical case as well as with respect to given ideals on $\omega $. We prove the following generalization of a result due to J. Brendle: If $V\subseteq W$ are transitive models, $\omega _1^W\subseteq V$, $\mathcal {P}(\omega )\cap V\not =\mathcal {P}(\omega )\cap W$, and $\mathcal {I}$ is an analytic or coanalytic ideal coded in $V$, then there is an $\mathcal {I}$-almost disjoint refinement of $\mathcal {I}^+\cap V$ in $W$. We study the existence of perfect $\mathcal {I}$-almost disjoint families, and the existence of $\mathcal {I}$-almost disjoint refinements in which any two distinct sets have finite intersection. We introduce the notion of mixing real (motivated by the construction of an almost disjoint refinement of $[\omega ]^\omega \cap V$ after adding a Cohen real to $V$) and discuss logical implications between the existence of mixing reals in forcing extensions and classical properties of forcing notions.

Détail

DOI : 10.4064/fm429-7-2017

Keywords: investigate families subsets omega almost disjoint refinements classical respect given ideals omega prove following generalization result due brendle subseteq transitive models omega subseteq mathcal omega cap mathcal omega cap mathcal analytic coanalytic ideal coded there mathcal almost disjoint refinement mathcal cap study existence perfect mathcal almost disjoint families existence mathcal almost disjoint refinements which distinct sets have finite intersection introduce notion mixing real motivated construction almost disjoint refinement omega omega cap after adding cohen real discuss logical implications between existence mixing reals forcing extensions classical properties forcing notions