How does set theory deal with Russell's antinomy, and antinomies generally? This is driving me crazy, i'm not too experienced with set Theory and formal logic, but what i am asking is that since we know, that if
$$R=\{X:X\not \in X\}$$
then $$R\in R\Leftrightarrow R \not \in R$$
But using this it is possible to prove any statement, how does one still let mathematics be rigourus, and talking about it, how does one know that in the end mathematics won't contradict itself? Are there other examples of such antinomies? If yes how does mathematics deal with them? Thanks for any answers, this really is driving me crazy.
 A: This is exactly why the axioms matter. We don't have a problem unless there is in fact a $R$ such that $R=\{x: x\not\in R\}$. So the way this - and everything else along the same lines - is handled is by carefully writing down a precise list of axioms we're going to use, and making sure that those axioms do not in fact allow this sort of nonsense.$^*$
There are a few candidate axiomatic frameworks floating around, but the current "common foundational system" is $\mathsf{ZFC}$. Roughly speaking, in $\mathsf{ZFC}$ we replace the naive "full comprehension" principle - namely that anything you can write between curly braces corresponds to a set - with a much more limited principle which only lets you build "small" sets. In particular, $\mathsf{ZFC}$ proves that for each set $A$ the set $$R_A=\{x\in A: x\not\in x\}$$ exists, but that's not a problem anymore!
Why not? Well, let's see what happens if we try to duplicate Russell's paradox here. As before we can argue $R_A\not\in R_A$. Now following Russell, our next step should be to deduce $R_A\in R_A$, which will be a contradiction. But now things are different because of the "$x\in A$" part of the definition of $R_A$: we can only conclude $R_A\in R_A$ if we've first proved both $R_A\not\in R_A$ and $R_A\in A$. So in fact what we get is just that $R_A$ is not in $R_A$ or $A$; there's no contradiction anymore.

$^*$What about potential other nonsenses we haven't thought of yet? Well, per Godel's theorem there's no hard and fast guarantee that our axioms are consistent; we can merely be "pretty sure," and take pains to understand which axioms are actually necessary for which results to prepare ourselves in case we need to change our axiom system in the event of a contradiction being discovered down the road. And hey, that task is pretty interesting in its own right.
