On the consistency of Borel's conjecture

For X a subset of [0, 1], there is a family of properties which X might have, each of which is stronger than X having Lebesgue measure zero, and each of which is trivially satisfied if X is countable.The main properties in this family (apart from the Lusin-type conditions, which are really meant to be stated in conjunction with 2 s, =~r or with Martin's axiom--see section 1) are X has strong measure zero, i.e., if e0, el ..... en .... (n <o)) is a sequence of positive reals then there exists a sequence I 0, 11, ..., I~ .... (n