A paradox I came up with, proof verification. I am wondering if the following is right? This is my solution to the following paradox.$$R = \{x : x \text{ not within }x,\,\text{where }x\text{ is a set}\}.$$Since by the nature of Russell's argument, we get a contradiction from the fact that if $R$ didn't contain $R$, then $R$ within $R$, as $R$ is not within $R$ from our assumption. Alternatively, if $R$ was in $R$ then we'd get a contradiction regarding the definition of $R$.
So my simple solution is that $R$ cannot exist. Since the contradiction arises from the earlier premise, that such a set of sets that do not contain themselves even exists. We only assumed (the one premise), the existence of $R$ in our argument.
This ultimately amounts to the idea that there cannot be a set of all sets or a set of all sets that do not contain themselves. The key word is all, which causes problems when you don't constrain a set to being able to handle lots of different sets. What I'm primarily referring to is the continuous power-setting of sets that would keep needing to be added after each insertion of a set into the set of all sets.
What does math.stackexchange think?
 A: Short version: yes, you are basically right. Russell's paradox can be construed as a theorem, that the full axiom of comprehension (for each formula-with-parameters, there is a set of all sets satisfying that formula) is inconsistent.
Your "continuous power-setting" idea sounds a lot like the cumulative hierarchy, the idea behind the set theory ZFC.

Note that there is a subtlety here: we can have a universal set, as long as separation (a local version of comprehension - for each formula-with-parameters and each set $A$, there is a set of all elements of $A$ satisfying the formula) doesn't always work. This is the situation with set theories like NFU, where comprehension (and hence separation) are restricted to formulas of a certain syntactic form, and "$x\not\in x$" is not one of them.
A: The contradiction arises in an early attempt to axiomatize set theory in the early 20th century. Using that set theory, you could both prove that $R$ does exist (by unrestricted comprehension), and, as you did here, that it cannot exist. Thus two contradictory theorems could be derived from that set theory -- not a good thing.
Various patches have been proposed over the years, the most widely used (by set theorists anyway) being the ZFC axioms for set theory.  
