Matrix iterations and Cichon's diagram

TL;DR

This paper demonstrates the consistency of assigning arbitrary regular cardinal values to various parts of Cichon's diagram using matrix iterations of ccc posets, expanding the understanding of possible cardinal configurations.

Contribution

It introduces new methods combining matrix iterations and previous techniques to control cardinal invariants in Cichon's diagram under ZFC.

Findings

Consistency of certain cardinal configurations in Cichon's diagram established.

Multiple arbitrary regular values can be assigned simultaneously.

New techniques expand the set of known possible cardinal arrangements.

Abstract

Using matrix iterations of ccc posets, we prove the consistency with ZFC of some cases where the cardinals on the right hand side of Cichon's diagram take two or three arbitrary values (two regular values, the third one with uncountable cofinality). Also, mixing this with the techniques in J Symb Log 56(3):795-810, 1991, we can prove that it is consistent with ZFC to assign, at the same time, several arbitrary regular values on the left hand side of Cichon's diagram.