Mathias--Prikry and Laver type forcing; Summable ideals, coideals, and $+$-selective filters

TL;DR

This paper investigates properties of Mathias--Prikry and Laver type forcings linked to filters and coideals, revealing their combinatorial characteristics, mutual embeddability, and effects on real numbers in set-theoretic extensions.

Contribution

It identifies key combinatorial properties of Mathias reals, classifies when Mathias forcing adds dominating or eventually different reals, and characterizes filters and families preserved under Laver type forcing.

Findings

Mathias--Prikry forcings with summable ideals are mutually bi-embeddable.

Mathias forcing with the complement of an analytic ideal adds a dominating real.

Certain filters are characterized by their behavior under forcing, especially regarding adding reals.

Abstract

We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias--Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias--Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of $\omega$-hitting and $\omega$-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.