The covering number of the strong measure zero ideal can be above almost everything else

Abstract

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal \({{\mathcal {S}}}{{\mathcal {N}}}\). As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that \(\mathrm {non}({{\mathcal {S}}}{{\mathcal {N}}})<\mathrm {cov}({{\mathcal {S}}}{{\mathcal {N}}})<\mathrm {cof}({{\mathcal {S}}}{{\mathcal {N}}})\), which is the first consistency result where more than two cardinal invariants associated with \({{\mathcal {S}}}{{\mathcal {N}}}\) are pairwise different. Another consequence is that \({{\mathcal {S}}}{{\mathcal {N}}}\subseteq s^0\) in ZFC where \(s^0\) denotes Marczewski’s ideal.