What is the relationship between the Hausdorff dimension and cardinality of a set?


Specifically, assuming the Continuum Hypothesis, if a set has Hausdorff dimension greater than zero does, that imply that its cardinality is equal too or greater than that of $2^{\aleph\_0}$?


Or, does the negation of CH, imply the existence of a set with positive Hausdorff dimension and cardinality strictly between $\aleph\_0$ and $2^{\aleph\_0}$?



Hausdorff dimension vs. cardinality
As stated, countable sets have Hausdorff dimension 0.
So any set $S$ with $\mathrm{HD}(S)>0$ has power $\ge \aleph\_1$.
No need for continuum hypothesis.


Without CH, though, we cannot say whether power $ \ge c = 2^{\aleph\_0}$ is required. But this is not about Hausdorff dimension, it is the same question for positive Lebesgue measure in the line. It is consistent with ZFC (follows from Martin's Axiom) that any set with power $< c$ has Lebesgue measure zero, or (for the same reason, or with the same proof, or even consequently) any set with power $< c$ has Hausdorff dimension zero. However, without CH (and without Martin's Axiom) there could be sets of reals of power $< c$ but with positive outer Lebesgue measure, and thus Hausdorff dimension 1.


