And if yes, then how can it be resolved?
As far as I know in standard set theory it's true that "no set is its own member". Also in standard logic the law of excluded middle is true, either $A$ or $\lnot A$.
Now let's consider simple sentence "Entity $X$ is either a real number or not a real number". This can be restated in terms of set theory as "Element $x$ either belongs to set $\mathbb{R}$ or to set non-$\mathbb{R}$".
Looks like a tautology, doesn't it? But I will show you that it's not because there is one thing that isn't a member of either set. Namely, it's set "non-$\mathbb{R}$". This can't belong to set $\mathbb{R}$ because it isn't a real number. This also can't belong to set non-$\mathbb{R}$ (i.e. it can't belong to itself) because of "no set is its own member". Thus "Element $x$ either belongs to set $\mathbb{R}$ or to set non-$\mathbb{R}$" isn't a tautology.
This is similar to Russell's paradox.
The usual approach is to cast blame on the principle of unrestricted comprehension — the hypothesis that for any property $P$ of sets, there is a set of everything satisfying $P$.
ZFC replaces this with the more modest hypothesis that if you're additionally given a set $S$, then you can form the set of everything in $S$ satisfying $P$.
In particular, in the usual formulation of modern set theory, the thing you call "non-R" is not a set. (c.f. "proper class")