The Higher Cicho\'n Diagram

TL;DR

This paper explores the relationships and inequalities between various ideals and their cardinal characteristics at a strongly inaccessible cardinal, revealing new phenomena and constructing models to distinguish these characteristics.

Contribution

It generalizes classical results to higher cardinals, introduces new concepts like stationary sets in this context, and demonstrates surprising inequalities and consistency results.

Findings

Some classical inequalities do not hold at higher cardinals.

The inequality cov(null) ≤ non(null) is established, contrary to classical expectations.

Models are constructed to distinguish cardinal characteristics using advanced forcing techniques.

Abstract

For a strongly inacessible cardinal $\kappa$, we investigate the relationships between the following ideals: - the ideal of meager sets in the ${<}\kappa$-box product topology - the ideal of "null" sets in the sense of [Sh:1004] (arXiv:1202.5799) - the ideal of nowhere stationary subsets of a (naturally defined) stationary set $S_{\rm pr}^\kappa \subseteq \kappa$. In particular, we analyse the provable inequalities between the cardinal characteristics for these ideals, and we give consistency results showing that certain inequalities are unprovable. While some results from the classical case ($\kappa=\omega$) can be easily generalized to our setting, some key results (such as a Fubini property for the ideal of null sets) do not hold; this leads to the surprising inequality cov(null)$\le$non(null). Also, concepts that did not exist in the classical case (in particular, the notion of stationary sets) will turn out to be relevant. We construct several models to distinguish the various cardinal characteristics; the main tools are iterations with $\mathord<\kappa$-support (and a strong "Knaster" version of $\kappa^+$-cc) and one iteration with ${\le}\kappa$-support (and a version of $\kappa$-properness).