So you are saying that supposing we had a set S with points/elements (rational numbers) such as {1.1, 1.2, 1.3, ...} - gotten from a real number line, we could pick any point, say 1.1 and scale/magnify it down by 0.1, and correspondingly we have points which is also a set, given below as
X = { 0.11, 0.22, 0.33,...., 1.1}
X has cardinality of 10
Which means that every point in S when scaled/magnified by a non-zero factor gives a new set of points ( e.g X for 1.1), and every point in this new set also gives it's own corresponding new set when also magnified, and so those the process continues to ad infinitum. The self-similarity here is that points are contained in a point when magnified. However, a single point has a topological dimension of zero which is equivalent to the point's fractal dimension but the fractal dimension (box-counting dimension) of any magnified point, say 1.1 in S above is given by
D = log(N)/log(1/e)
D = fractal/box-counting dimension
N = number of points in the magnified point (1.1)
e = scale/magnification factor.
In our scenario as seen above
N = 10 (cardinality of set x)
e = 0.1
Therefore D = 1
This seems to me like a fractal.
It works more general for all the rationals. I think it is true that any subset of the rationals gives a box counting dimension \leq 1
The main point is that the Hausdorf dimension is zero. Because it is countable. Thefore, the Hausdorf dimension says it is not a fractal whereas the Boxcounting dimensions says it is a fractal.
Take it easy comrade, I'm aware
It (subset of rationals) actually does.
I think I get where you are coming from.
It's nice learning from you anyways, thanks.
I am happy to share :3
Ah yes, this is obviously true :P
An interesting exercise would be to construct a non-finite subset of the rationals with box counting dimension equal to zero :3
Ok, good luck with that.