Almost disjoint refinements and mixing reals

TL;DR

This paper explores the structure and properties of almost disjoint refinements and mixing reals within set theory, focusing on ideals on natural numbers, their refinements, and implications for forcing and models.

Contribution

It generalizes a result on almost disjoint refinements for analytic ideals, constructs perfect and special ideals with unique properties, and links forcing notions to mixing reals.

Findings

Existence of $ ext{ADR}$ for certain ideals under model extensions.

Construction of perfect $ ext{AD}$ families and special ideals.

Connections established between forcing properties and mixing reals.

Abstract

We investigate families of subsets of $\omega$ with almost disjoint refinements in the classical case as well as with respect to given ideals on $\omega$. More precisely, we study the following topics and questions: 1) Examples of projective ideals. 2) 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 ($\mathcal{I}$-ADR) of $\mathcal{I}^+\cap V$ in $W$, that is, a family $\{A_X:X\in\mathcal{I}^+\cap V\}\in W$ such that (i) $A_X\subseteq X$, $A_X\in \mathcal{I}^+$ for every $X$ and (ii) $A_X\cap A_Y\in\mathcal{I}$ for every distinct $X$ and $Y$. 3) The existence of perfect $\mathcal{I}$-almost disjoint ($\mathcal{I}$-AD) families, and the existence of a "nice" ideal $\mathcal{I}$ on $\omega$ with the property: Every $\mathcal{I}$-AD family is countable but $\mathcal{I}$ is nowhere maximal. 4) The existence of $(\mathcal{I},\text{Fin})$-almost disjoint refinements of families of $\mathcal{I}$-positive sets in the case of everywhere meager (e.g. analytic or coanalytic) ideals. We prove a positive result under Martin's Axiom. 5) Connections between classical properties of forcing notions and adding mixing reals (and mixing injections), that is, a (one-to-one) function $f:\omega\to\omega$ such that $|f[X]\cap Y|=\omega$ for every $X,Y\in [\omega]^\omega\cap V$.