Strong measure zero sets without Cohen reals

Abstract

Abstract If ZFC is consistent, then each of the following is consistent with : (1) X ⊆ ℝ is of strong measure zero iff ∣ X ∣ ≤ ℵ 1 + there is a generalized Sierpinski set. (2) The union of ℵ many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size ℵ 2 + there is no Cohen real over L .