The Point-to-Set Principle, the Continuum Hypothesis, and the Dimensions of Hamel Bases
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We prove that the Continuum Hypothesis implies that every real number in (0,1] is the Hausdorff dimension of a Hamel basis of the vector space of reals over the field of rationals. 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).
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