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?

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 $2^{\aleph_0}$?

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?

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?

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 the Continuum?

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?