Cardinal invariants of idealized Miller null sets

TL;DR

This paper investigates the cardinal invariants of a class of $\sigma$-ideals related to Miller forcing, computing their properties for various ideals and extending Cichoń's Maximum with new invariants.

Contribution

It introduces and analyzes the $\mathscr{I}$-Miller null ideals, computing their cardinal invariants and connecting them to Cichoń's Maximum, expanding understanding of idealized forcing.

Findings

Computed cardinal invariants for typical $\mathscr{I}$-Miller null ideals.

Extended Cichoń's Maximum with invariants from these ideals.

Established relationships between $\mathscr{I}$-Miller ideals and classical forcing notions.

Abstract

This paper provides an extensive study of the $\mathscr{I}$-Miller null ideals $M_\mathscr{I}$, $\sigma$-ideals on the Baire space parametrized by ideals $\mathscr{I}$ on countable sets. These $\sigma$-ideals are associated to the idealized versions of Miller forcing in the same way that the meager ideal is associated to Cohen forcing. We compute the cardinal invariants of $M_\mathscr{I}$ for typical examples of Borel ideals $\mathscr{I}$ and show that Cicho\'{n}'s Maximum can be extended by adding the uniformity and covering numbers of $M_\mathscr{I}$ for different ideals $\mathscr{I}$.