Mathias and Silver forcing parametrized by density

TL;DR

This paper explores variants of Silver and Mathias forcing based on density measures, analyzing their properties and effects on set-theoretic structures, including properness, chain conditions, and regularity properties.

Contribution

It introduces density-parametrized Silver and Mathias forcing notions and examines their impact on properness, chain conditions, and regularity properties, revealing both collapsing and gentle forcing behaviors.

Findings

Some forcings collapse 2^ω to ω

Others preserve cardinals and are gentle

Connections to Baire property are established

Abstract

We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse 2^\omega to \omega, while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property.