We study the following natural strong variant of destroying Borel ideals: $\mathbb {P}$ $+$-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

$$ \begin{align*} \mathrm{non}^*(\mathcal{I},+)=&\min\big\{|\mathcal{Y}|:\mathcal{Y}\subseteq\mathcal{I}^+,\; \forall\;A\in\mathcal{I}\;\exists\;Y\in\mathcal{Y}\;|A\cap Y|<\omega\big\},\\ \mathrm{cov}^*(\mathcal{I},+)=&\min\big\{|\mathcal{C}|:\mathcal{C}\subseteq\mathcal{I},\; \forall\;Y\in\mathcal{I}^+\;\exists\;C\in\mathcal{C}\;|Y\cap C|=\omega\big\} \end{align*} $$

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 $\mathcal {I}$; and (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal.

We investigate which filters on ω can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in ${[\omega ]^\omega }$). We prove the following results:

1. (1) Many classical examples of nice tall filters contain no towers (in ZFC).

2. (2) It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well).

3. (3) It is consistent that all towers generate nonmeager filters (this answers a question of P. Borodulin-Nadzieja and D. Chodounský), in particular (consistently) Borel filters do not contain towers.

4. (4) The statement “Every ultrafilter contains towers.” is independent of ZFC (this improves an older result of K. Kunen, J. van Mill, and C. F. Mills).

Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters (${\rm{ad}}{{\rm{d}}^{\rm{*}}}\left( {\cal F} \right)$, ${\rm{co}}{{\rm{f}}^{\rm{*}}}\left( {\cal F} \right)$, ${\rm{no}}{{\rm{n}}^{\rm{*}}}\left( {\cal F} \right)$, and ${\rm{co}}{{\rm{v}}^{\rm{*}}}\left( {\cal F} \right)$), and the existence of Luzin type families (of size $\ge {\omega _2}$), that is, if ${\cal F}$ is a filter then ${\cal X} \subseteq {[\omega ]^\omega }$ is an ${\cal F}$-Luzin family if $\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$ is countable for every $F \in {\cal F}$.