Do the Zermelo-Fraenkel axioms prevent the existence of fractals?

Question

It is well known that Zermelo-Fraenkel axioms solve the Russell's paradox - i.e. A set can not contain itself (singular set) without leading to logical contradictions- just saying that this kind of sets can not exist. On the other hand we have chaos theory which is based on self-similar sets called fractals that, at least, seems to exist because it governs the dynamics of non-linear systems. In this context, are the ZF axioms and the existence of fractals in some sense contradictory?

Answer 1

No, for the same reason that it doesn't prevent the reals, or rationals, or even just the integers, from existing. After all, there is also an axiom that implies there is no infinite descending sequence of elements, so how can $\Bbb Z$ have no minimal element?

The answer is simple, we can endow a set with structure that has nothing to do with $\in$.

Indeed, if you think about "self-similarity" as being isomorphic to a proper subset, then to some extent, that is how we define infinite sets (under the assumption of the Axiom of Choice, anyway). Indeed, $A$ is infinite, if and only if there is $B\subsetneq A$ and a bijection $f\colon A\to B$. In the category of sets, where no structure is given, a bijection is an isomorphism. And so, to some extent, without structure all infinite sets are "fractals".

Fractals, generally, are defined in a context of geometry, or at least some sort of a topology is necessary. That's a whole other structure that does not need to abide to the axioms of $\sf ZFC$.