If ZFC is consistent, then each of the following is consistent with ZFC + 2ℵ0 = ℵ2: (1) $X \subseteq \mathbb{R}$ is of strong measure zero iff |X| ≤ ℵ1 + there is a generalized Sierpinski set. (2) The union of ℵ1 many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size ℵ2 + there is no Cohen real over L.