Mathias and silver forcing parametrized by density

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? to ?, while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property.

Automated summary

This paper introduces and investigates versions of Silver and Mathias forcing with respect to lower and upper density. The focus is on properness, Axiom A, chain conditions, preservation of cardinals, and adding Cohen reals. We discover rough forcings that collapse 2ᶿ to ᶿ, while others are surprisingly gentle. We also study the connections between regularity properties induced by these parametrized forcing notions and the Baire property.