Almost disjoint refinements and mixing reals
Volume 242 / 2018
Abstract
We investigate families of subsets of 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.
