WAYS OF DESTRUCTION

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. \nAmong 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.