Abstract: We prove that, for every 0 ⩽ s ⩽ 1 , there is a Hamel basis of the vector space of reals over the field of rationals that has Hausdorff dimension s . The logic of our proof is of particular interest. The statement of our theorem is classical; it does not involve the theory of computing. However, our proof makes essential use of algorithmic fractal dimension–a computability-theoretic construct–and the point-to-set principle of J. Lutz and N. Lutz (2018).