I can't understand Russell's paradox. What I understand is that Russell's paradox arises because the set of all sets that are members of themselves is empty. That it's impossible to find a set that's a member of itself, but one can define the set of all sets of the universe that clearly contain itself. Does it mean that there is no set of all sets of the universe?
Please, make answers as simple as possible, I'm nearly ignorant in set theory.
Russell's paradox, as well other paradoxes (Cantor's paradox, Burali-Forti paradox) simply tell us that some collections that we can define are not sets. There are two ways to overcome these things:
In modern set theory such as ZFC (Zermelo-Fraenkel with Choice) the collection of all sets is not a set. This is why there is a notion of class, it simply means a collection which we can define. In the case of "all the sets" we just define it to be $\{x\mid x=x\}$.
There are, however, set theories in which the collection of all sets is a set. One example is NF (New Foundations). In this theory we limit the formulas which define new sets, and so Russell's collection cannot become a set. This means that we cannot prove from NF that the collection defined by Russell's paradox is a set itself, so no contradiction arises.
Further reading material:
The fact that (in ZFC) a set cannot be a member of itself has no real connection to the Russell Paradox. It is a consequence of the Axiom of Regularity..
The Axiom of Regularity was introduced by von Neumann for purely technical reasons that had nothing to do with paradox avoidance. For one thing, it makes the development of the theory of ordinals smoother. However, set theory can be perfectly well developed without Regularity.
The axioms of ZFC are intended to be (i) powerful enough to allow all the set constructions that we need for a mathematics based on Set Theory and (ii) not so powerful as to lead to a contradiction.
One cannot know for sure that inconsistency has been avoided. But the cheap kind of inconsistency that doomed the logic of Frege is, one hopes, made impossible by putting limitations on how one is allowed to construct sets. (Very) roughly speaking, one is allowed to construct a set from the contents of an already constructed set (here the major exception is the Axiom of Infinity, along with parts of the Axiom scheme of Replacement).
In case you want a short answer that does not attempt to add too much perspective:
Russell's paradox does not prove or require that $\{x\mid x\in x\}$ is empty. It proves that $\{x\mid x\not\in x \}$ cannot exist as a set (which is different from being empty -- an empty set does exist), no matter whether there are sets that contain themselves or not.
In ordinary ZFC set theory there is indeed no "set of all sets of the universe". (But see Asaf's answer) .
In ordinary ZFC set theory $\{x\mid x\in x\}$ is indeed the empty set -- and therefore $\{x\mid x\in x\}$ happens to exist there. (But see André's answer).
I have had a problem understanding this before. Here's what I can provide.
The Russell's paradox says that it is impossible to define the set $U$ of all things if $U$ is also considered one of "all things". By impossible, I mean if you allow $U$ to exist, you will violate one of many intuitive rules. You may, as a result, convince yourself that maybe some "intuitive" rules are not supposed to hold as a way to resolve Russell's paradox.
A property $x \notin x$ is considerably nice, so intuitively it should not pose a problem if I pick a subset $V \subseteq U$ by picking elements that satisfy $x \notin x$. However, Russell's paradox says that if you define $V$ like this, you will not know whether $V \in V$ or $V \notin V$. This is considered a contradiction to our basic assumption that you must be able to tell whether $x \in y$ or $x \notin y$ if $y$ is a set.
There are several ways to resolve Russell's paradox. I will list some here.
Anyway, the root of all evil is self-referencing. I feel that it's a common theme in all branches of mathematics.
