This appears unanswered on reddit.
What are possible choices for the cardinality of plane or line fractal without assuming CH?
Are there fractals for which the answer is not easy? (Sieprinski triangle is open on reddit.)
This appears unanswered on reddit.
What are possible choices for the cardinality of plane or line fractal without assuming CH?
Are there fractals for which the answer is not easy? (Sieprinski triangle is open on reddit.)
This question appears to be off-topic. The users who voted to close gave this specific reason:
Let's define that a subset $F$ of the plane (or other suitable space) is a fractal, if there is a nontrivial group $G$ of homeomorphisms of the space, at least some of which are not isometries, such that $F=\sigma F$ for every $\sigma\in G$. This expresses the self-similarity feature of fractals.
With this definition, it is easy to find fractals $F$ of any desired infinite cardinality less than or equal to the continuum. Indeed, for any group $G$, there will be a fractal $F$ with respect to $G$ of any size between $\max(|G|,\aleph_0)$ and $\frak{c}$, inclusive.
To build $F$, just pick any collection of starting points in the plane, of the right cardinality, and then simply close the set under all the operations of $G$. This closure will have the desired cardinality, and it follows that $F=\sigma F$ for every $\sigma\in G$, and so $F$ is a fractal.
In particular, the procedure produces countable fractals $F$ with respect to any fixed countable group $G$. One simply starts with a single point $\{a\}$ and then closes under the $G$ action.
One can get versions of the Sierpinski triangle this way, by using the group $G$ that describes the fractal symmetry of that triangle.
I might add that this is a common way of drawing fractals on the computer. One just picks a point and then draws all its images, and then their images and so on, under the desired symmetries.
Often, one doesn't want to use a whole group of homeomorphisms, but a weaker collection of maps that exhibit the desired self-similarities, such as the contractive maps only, but the same idea works for this case. For example, with the fern above and the Sierpinski triangle, one doesn't usually apply the expansive maps, and so the resulting image is bounded. But even in those cases, one could apply the inverse maps that produce an unbounded set exhibiting the whole group of symmetries. In this way, one gets a more expansive version of the fern or the Sierpinski triangle that continues outward, exhibiting the desired similarities more fully.