HL ideals and Sacks indestructible ultrafilters

TL;DR

This paper explores ultrafilters and reaping families that remain intact under Sacks forcing, providing combinatorial characterizations and demonstrating the indestructibility of certain families and ultrafilters.

Contribution

It offers a new combinatorial characterization of Sacks indestructible families and ultrafilters, showing their existence and properties in relation to definable ideals.

Findings

Every reaping family smaller than continuum is Sacks indestructible.

Complements of many definable ideals are Sacks reaping indestructible.

Every Sacks indestructible ultrafilter is a z-ultrafilter.

Abstract

We study ultrafilters on countable sets and reaping families which are indestructible by Sacks forcing. We deal with the combinatorial characterization of such families and we prove that every reaping family of size smaller than the continuum is Sacks indestructible. We prove that complements of many definable ideals are Sacks reaping indestructible, with one notable exception, the complement of the ideal $\mathcal Z$ of sets of asymptotic density zero. We investigate the existence of Sacks indestructible ultrafilters and prove that every Sacks indestructible ultrafilter is a $\mathcal Z$-ultrafilter.