Ways of Destruction

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*} $\mathrm{n}\mathrm{o}\mathrm{n}$^($\mathcal{I}$,+)=&\min\big{|$\mathcal{Y}$|:$\mathcal{Y}$\subseteq$\mathcal{I}$^+,; \forall;A\in$\mathcal{I}$;\exists;Y\in$\mathcal{Y}$;|A\cap Y|<\omega\big}\ $\mathrm{c}\mathrm{o}\mathrm{v}$^($\mathcal{I}$,+)=&\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, $\mathbb{M}({\mathcal{I}}^{\ast })$-generic real $+$-destroys $\mathcal{I}$ iff $\mathbb{M}({\mathcal{I}}^{\ast })$ $+$-destroys $\mathcal{I}$ iff $\mathcal{I}$ can be $+$-destroyed iff ${\mathrm{c}\mathrm{o}\mathrm{v}}^{\ast }(\mathcal{I},+)>\omega$; (4) we characterise when the Laver-Prikry, $\mathbb{L}({\mathcal{I}}^{\ast })$-generic real $+$-destroys $\mathcal{I}$, and in the case of P-ideals, when exactly $\mathbb{L}({\mathcal{I}}^{\ast })$ $+$-destroys $\mathcal{I}$; (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal.