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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?

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    $\begingroup$ ”Self-similarity” in the context of dynamical systems does not mean that a “set contains itself”. $\endgroup$
    – Martin R
    Sep 23, 2021 at 7:52
  • $\begingroup$ Ouch, I thought it was. Can you give me a more detailed explanation please? $\endgroup$
    – T. ssP
    Sep 23, 2021 at 7:55
  • $\begingroup$ en.wikipedia.org/wiki/Self-similarity#Definition $\endgroup$
    – Martin R
    Sep 23, 2021 at 7:58
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    $\begingroup$ Straight line is also a self-similar set, and ZF axioms sure do not prevent its existence. Self-similarity just means that any open interval of the line, no matter how small, is homeomorphic to the whole line. I.e. there is a continuous map from one to the other with continuous inverse. $\endgroup$
    – Conifold
    Sep 23, 2021 at 8:15
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    $\begingroup$ ZFC does not solve Russel's paradox by contradicting the existence of a set containing itself: the only consequence, when defining the set containing all sets which don't contain themselves, would be that this set is in fact all the sets. But then it would contain itself, and we would have a contradiction, hence it would still be a paradox. What removes this paradox in ZFC is the fact that we cannot prove that such a set exists (probably). In ZFC, not everything is a set, and we are limited in the sets we can define. We cannot define the sets of all sets, or the sets of all ordinals, etc. $\endgroup$ Sep 23, 2021 at 8:28

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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 $\sf ZFC$.

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