Controlling classical cardinal characteristics while collapsing cardinals

TL;DR

This paper demonstrates how to manipulate forcing notions to control classical cardinal characteristics of the reals while collapsing cardinals, and how to assign distinct values to certain invariants using Boolean ultrapower techniques.

Contribution

It introduces methods to preserve and control cardinal characteristics during forcing extensions, including collapsing cardinals and separating invariants like , , and  in Cichob4n's diagram.

Findings

Successfully collapses cardinals while maintaining specified cardinal characteristics.

Forces distinct values for , , and  invariants.

Uses Boolean ultrapower method to achieve these results.

Abstract

Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $\lambda>\kappa$ to $\kappa$) such that the composition still forces the previous values to these characteristics. We also show how to force distinct values to $\mathfrak m$, $\mathfrak p$ and $\mathfrak h$ and also keeping all the values in Cicho\'n's diagram distint, using the Boolean Ultrapower method of arXiv:1708.03691 . (In arXiv:2006.09826 , the same was done for the newer Cicho\'n's Maximum construction, which avoids large cardinals.)