The part of the question about the continuum hypothesis (CH) seems confused: without assuming (CH) (but assuming axiom of choice so that cardinals work as they should), aleph_1 is by definition the least uncountable cardinal. (The continuum hypothesis asserts that aleph_1 = 2^{aleph_0}.)

Let X be a metric space, and let x in X. Then it follows immediately from the definition -- see e.g.

http://en.wikipedia.org/wiki/Hausdorff_measure

that for any d > 0, the d-dimensional Hausdorff measure H_d({x}) is equal to zero. (This is just because a point can be covered by a single ball with arbitrarily small diameter.) Since H_d is a measure, it is countably additive: thus H_d(S) = 0 for any countable set S. If H_d(S) = 0, then the Hausdorff dimension of S is at most d, so H_d(S) = 0 for all
positive d implies that the Hausdorff dimension of d = 0.

Possibly you wanted to ask: without assuming CH, does a set of positive Hausdorff dimension necessarily have at least continuum cardinality? (I don't know the answer.)