We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal 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 non(SN ) < cov(SN ) < cof(SN ), which is the first consistency result where more than two cardinal invariants associated with SN are pairwise different. Another consequence is that SN ⊆ s 0 in ZFC where s 0 denotes Marczewski's ideal.

By coding Polish metric spaces with metrics on countable sets, we propose an interpretation of Polish metric spaces in models of ZFC and extend Mostowski's classical theorem of absoluteness of analytic sets for any Polish metric space in general. In addition, we prove a general version of Shoenfield's absoluteness theorem.

The results of characterizing the alumina ball size distribution in two mills of a crushing and grinding plant are shown. The mills were unloaded and the ball charge was screened in order to establish the ball size distribution. For both mills, the balls retained during the unloading were compared to the balls retained at the beginning of the process, and additionally, they were compared to the results obtained by the Swebrec adjusted distribution model. In both cases, the experimental data have had a good fit with this model. This practice is important in order to establish the best ball charge at the beginning of the operation and the ball recharge in the steady state.

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal 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 non(SN ) < cov(SN ) < cof(SN ), which is the first consistency result where more than two cardinal invariants associated with SN are pairwise different. Another consequence is that SN ⊆ s 0 in ZFC where s 0 denotes the Marczewski's ideal.