强测度零理想的覆盖数几乎可以高于其它所有的覆盖数
The covering number of the strong measure zero ideal can be above almost everything else
摘要：
我们发现，某些类型的树强迫，包括萨克斯强迫，增加覆盖的强措施零理想$\mathcal{SN}$。因此，在Sacks模型中，这样的覆盖数等于连续统的大小，这表明这个覆盖数始终大于连续统的任何其他经典基数不变量.甚至，可以使用Sacks强制来强制该$\mathrm{non}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$，这是与$\mathcal{SN}$相关联的两个以上基数不变量成对不同的第一一致性结果。另一个结果是ZFC中的$\mathcal{SN}\subseteq s^0$，其中$s^0$表示Marczewski理想。
英文摘要：
We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal $\mathcal{SN}$. 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 $\mathrm{non}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$, which is the first consistency result where more than two cardinal invariants associated with $\mathcal{SN}$ are pairwise different. Another consequence is that $\mathcal{SN}\subseteq s^0$ in ZFC where $s^0$ denotes the Marczewski's ideal.
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