Let $\mathcal{ SN}$ be the $\sigma$-ideal of the strong measure zero sets of reals. We present general properties of forcing notions that allow to control of the additivity of $\mathcal{ SN}$ after finite support iterations. This is applied to force that the four cardinal characteristics associated with $\mathcal{ SN}$ are pairwise different: \[\mathrm{ add}(\mathcal{ SN})<\mathrm{ cov}(\mathcal{ SN})<\mathrm{ non}(\mathcal{ SN})<\mathrm{ cof}(\mathcal{ SN}).\] Furthermore, we construct a forcing extension satisfying the above and Cicho\'n's maximum (i.e.\ that the non-dependent values in Cicho\'n's diagram are pairwise different).
Categories: math.LO