Separating cardinal characteristics of the strong measure zero ideal

Abstract

Let $\mathcal{S}\mathcal{N}$ 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 $\mathcal{S}\mathcal{N}$ after finite support iterations. This is applied to force that the four cardinal characteristics associated with $\mathcal{S}\mathcal{N}$ 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 Cichoń's maximum (i.e.\ that the non-dependent values in Cichoń's diagram are pairwise different).

Final version

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