Almost disjoint refinements and mixing reals

作者：Farkas Barnabas^*; Khomskii Yurii; Vidnyanszky Zoltan

来源：Fundamenta Mathematicae, 2018, 242(1): 25-48.

DOI：10.4064/fm429-7-2017

• analytic ideal
• coanalytic ideal
• almost disjoint family
• almost disjoint refinement
• Mansfield Solovay theorem
• mixing real
• meager ideal

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 subset of W are transitive models, omega(W)(1) subset of V, P(omega) boolean AND V not equal P(omega) boolean AND W, and J is an analytic or coanalytic ideal coded in V, then there is an J-almost disjoint refinement of J(+) boolean AND V in W.
We study the existence of perfect J-almost disjoint families, and the existence of J-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) boolean AND 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.

• 出版日期2018