Does not the existence of fractals with fractional Hausdorff dimension prove that there are cardinalities in between countable and continuum?

Question

Is not it the case that the cardinality of fractals of dimension 0.00001 should be greater than countable but less than that of continuum?

Answer 1

No, it does not.

The underlying assumption that dimension correlates strongly with cardinality is false. Indeed, this is one of the earliest results about infinite sets: that (for example) $\mathbb{R}$ and $\mathbb{R}^2$ have the same cardinality despite their different dimensions. Note that fractals don't enter into this at all: right at the outset we see a fundamental difference betwen the "geometric" characteristics of a particular set of points and the cardinality of that set.

As an exercise, it's not hard to construct a bijection between the Koch curve and the unit interval. In fact, on general descriptive-set-theoretic grounds no "reasonably-definable" shape is going to have cardinality strictly between $\aleph_0$ and $2^{\aleph_0}$ (e.g. no continuous image of a Borel set will do this), regardless of whether CH holds. So all the fractals we can actually describe concretely have size continuum.

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There is a related question, however, which is more interesting:

Is it the case that every set of positive fractal dimension has size continuum?

It turns out that this is not answerable by ZFC alone. Specifically, if we let $\mathfrak{h}$ be the smallest cardinality of any subset of $\mathbb{R}$ of positive Hausdorff dimension (say), it turns out that:

• It is consistent with ZFC that $\mathfrak{h}=2^{\aleph_0}$. (This happens trivially if CH holds, but it's more interestingly also a consequence of the weaker principle MA which allows $2^{\aleph_0}>\aleph_1$.)

• It is consistent with ZFC that $\mathfrak{h}<2^{\aleph_0}$. (This requires forcing.)

Meanwhile, ZFC obviously proves that $\aleph_0<\mathfrak{h}\le 2^{\aleph_0}$. Quantities like $\mathfrak{h}$ - reasonably-definable cardinals corresponding to notions of "sufficiently large" which we know are uncountable and at most continuum, but consistently are intermediate - are called cardinal characteristics of the continuum, and their interplay is extremely well-studied (see e.g. here or here).