  Constructive dimension and constructive strong dimension are effectivizations
of the Hausdorff and packing dimensions, respectively. Each infinite binary
sequence A is assigned a dimension dim(A) in [0,1] and a strong dimension
Dim(A) in [0,1].
  Let DIM^alpha and DIMstr^alpha be the classes of all sequences of dimension
alpha and of strong dimension alpha, respectively. We show that DIM^0 is
properly Pi^0_2, and that for all Delta^0_2-computable alpha in (0,1],
DIM^alpha is properly Pi^0_3.
  To classify the strong dimension classes, we use a more powerful effective
Borel hierarchy where a co-enumerable predicate is used rather than a
enumerable predicate in the definition of the Sigma^0_1 level. For all
Delta^0_2-computable alpha in [0,1), we show that DIMstr^alpha is properly in
the Pi^0_3 level of this hierarchy. We show that DIMstr^1 is properly in the
Pi^0_2 level of this hierarchy.
  We also prove that the class of Schnorr random sequences and the class of
computably random sequences are properly Pi^0_3.
