# Ways of Destruction

## TL;DR

This paper explores a strong variant of destroying Borel ideals using forcing, characterizes when such destruction occurs, and examines implications for classical ideals and related cardinal invariants.

## Contribution

It introduces a new strong destruction concept for Borel ideals, provides combinatorial criteria, and analyzes destruction by specific forcing notions like Mathias-Prikry and Laver-Prikry.

## Key findings

- Characterization of when a real forcing can $+$-destroy a Borel ideal
- Analysis of $+$-destructibility for classical Borel ideals
- Equivalence conditions for $+$-destruction by Mathias-Prikry forcing

## Abstract

We study the following natural strong variant of destroying Borel ideals: $\mathbb{P}$ $\textit{$+$-destroys}$ $\mathcal{I}$ if $\mathbb{P}$ adds an $\mathcal{I}$-positive set which has finite intersection with every $A\in\mathcal{I}\cap V$. Also, we discuss the associated variants \begin{align*} \mathrm{non}^*(\mathcal{I},+)=&\min\big\{|\mathcal{Y}|:\mathcal{Y}\subseteq\mathcal{I}^+,\; \forall\;A\in\mathcal{I}\;\exists\;Y\in\mathcal{Y}\;|A\cap Y|<\omega\big\}\\ \mathrm{cov}^*(\mathcal{I},+)=&\min\big\{|\mathcal{C}|:\mathcal{C}\subseteq\mathcal{I},\; \forall\;Y\in\mathcal{I}^+\;\exists\;C\in\mathcal{C}\;|Y\cap C|=\omega\big\} \end{align*} of the star-uniformity and the star-covering numbers of these ideals. Among other results, (1) we give a simple combinatorial characterisation when a real forcing $\mathbb{P}_I$ can $+$-destroy a Borel ideal $\mathcal{J}$; (2) we discuss many classical examples of Borel ideals, their $+$-destructibility, and cardinal invariants; (3) we show that the Mathias-Prikry, $\mathbb{M}(\mathcal{I}^*)$-generic real $+$-destroys $\mathcal{I}$ iff $\mathbb{M}(\mathcal{I}^*)$ $+$-destroys $\mathcal{I}$ iff $\mathcal{I}$ can be $+$-destroyed iff $\mathrm{cov}^*(\mathcal{I},+)>\omega$; (4) we characterise when the Laver-Prikry, $\mathbb{L}(\mathcal{I}^*)$-generic real $+$-destroys $\mathcal{I}$, and in the case of P-ideals, when exactly $\mathbb{L}(\mathcal{I}^*)$ $+$-destroys $\mathcal{I}$; (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal.