Cardinal invariants of idealized Miller null sets

TL;DR

The paper studies the cardinal invariants of the idealized Miller nulls $M_\mathscr{I}$ and their $K_\mathscr{I}$-counterparts on the Baire space, parameterized by ideals $\mathscr{I}$ on countable sets. It develops an exact framework tying the $\omega$-versions of star-numbers $\mathrm{add}^*_\omega(\mathscr{I})$, $\mathrm{non}^*_\omega(\mathscr{I})$, and $\mathrm{cof}^*_\omega(\mathscr{I})$ to the invariants of $M_\mathscr{I}$ and $K_\mathscr{I}$, and computes these values for key Borel ideals (e.g., $\mathrm{nwd}$, $\mathcal{S}$, $\mathcal{R}$, $\mathrm{conv}$, $\mathcal{ED}$, $\mathcal{ED}_{\mathrm{fin}}$, $\mathrm{Fin}\otimes\mathrm{Fin}$, $\mathcal{Z}$). The authors prove ZFC equalities like $\mathrm{add}(M_\mathscr{I})=\mathrm{add}(K_\mathscr{I})=\min\{\mathfrak{b},\mathrm{add}^*_\omega(\mathscr{I})\}$ and $\mathrm{cof}(M_\mathscr{I})=\mathrm{cof}(K_\mathscr{I})=\max\{\mathfrak{d},\mathrm{cof}^*_\omega(\mathscr{I})\}$, and relate uniformity/covering numbers to evasion/prediction variants. The paper then advances consistency results via forcing techniques—Fr-limits, closed-Fr-limits, and UF-limits—constructing models extending Cichoń's maximum to incorporate invariants of $M_\mathscr{I}$ and $K_\mathscr{I}$, including the introduction of new ideals $\mathscr{J}_L$ and $\mathscr{I}^L_f$ and associated forcings $\mathbb{P}^{\mathscr{J}_L}$ and $\mathbb{P}^{\mathscr{I}_L^f}$. These results demonstrate substantial separations between left and right sides of the diagram and show that idealized Miller null sets yield rich, controllable extensions of Cichoń's maximum with precise invariant values.

Abstract

This paper provides an extensive study of the $\mathscr{I}$-Miller null ideals $M_\mathscr{I}$, $σ$-ideals on the Baire space parametrized by ideals $\mathscr{I}$ on countable sets. These $σ$-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 Cichoń's Maximum can be extended by adding the uniformity and covering numbers of $M_\mathscr{I}$ for different ideals $\mathscr{I}$.

Figures (5)

• Figure 1: Cichoń's diagram. An arrow $\mathfrak{x}\to\mathfrak{y}$ denotes that $\mathfrak{x}\leq\mathfrak{y}$ holds.
• Figure 2: Extended Cichoń's Maximum. $f$ represents any function $f\in{\omega}^\omega$ such that $\lim_{n\to\infty}f(n)=\infty$ and $\lim_{n\to\infty}f(n)/2^n=0$.
• Figure 3: Diagram of Borel ideals and Katětov-Blass orders. An arrow $\mathscr{I}\to\mathscr{J}$ denotes that $\mathscr{I}\leq_{\mathrm{KB}}\mathscr{J}$ holds.
• Figure 4: Separation Constellation of \ref{['thm:CM_left']}. Note that $\mathop{\mathrm{non}}\nolimits(M_\mathcal{Z})=\max\{\mathfrak{b},\mathop{\mathrm{non}}\nolimits(\mathcal{E})\}$ by \ref{['thm:nonMZ_exact_value']}.
• Figure 5: Separation Constellation of \ref{['thm:CM']}.

Theorems & Definitions (258)

• Definition 1.1
• Definition 1.2
• Lemma 1.3: Sabok--Zapletal SZ11
• Theorem 1: \ref{['thm:exact_values_of_add']}, \ref{['lem:b_nonM']}, \ref{['thm:nonmi_leq_max']}
• Theorem 2
• Theorem 3
• Theorem 4: \ref{['thm:CM']}
• Definition 2.1
• Definition 2.2
• Definition 2.3