The version of Cantor's notion of sets that I've come across goes something like this:
"...collection of well defined, distinguishable objects of our intuition or of our thought to be conceived of as a whole. The objects are called the members of the set..."
With Russell's paradox $B = \{x:x \notin x\}$, I understand the mistake is assuming the collection $B$ is a set (i.e. if by sets we mean a collection which has the membership relation with its elements). The paradox shows not all collections are sets.
So far I haven't seen any paradox phrased like this: $B = \{x:x \notin B\}$? It seems slightly different from Russell's paradox in that the question isn't so much about whether $B$ is a collection which is also a set, but whether $B$ is a collection at all. Is this formulation allowed in Cantor's notion of sets, where it must satisfy the criterion of being well defined?
Thanks!
The basic problem here is that even in naive set theory you have no right to assume that $$ B = \{ x\mid x\notin B\} $$ works as the definition of $B$. Since the letter $B$ appears in the expression that supposedly defines its meaning, what you have is not a definition, but merely an equation that you want $B$ to be a solution of. And there's nothing paradoxical about the fact that this equation happens not to have any solution, any more than it is paradoxical that $$ x=x+1 $$ fails to define a number $x$.
Even if we look at the non-contradictory case, $$ C = \{ x \mid x \in C \} $$ that too fails to define anything because it just says that $C$ equals itself, which does not single out any particular set among all the other sets that also equal themselves. Again, this is not mysterious, but the same as the fact that $$ y = 1\cdot y $$ fails to define a number $y$.