Q:

How does Russell's paradox affect Cantor's set theory?

I've been studying naive set theory and I have been told that Russell's paradox causes problems in Cantor's set theory when sets get "too big". 
I don't understand why this causes a problem. I know how the paradox effected Frege's work in terms of logic but not Cantor. 
Any help will be appreciated, 
Thanks

A:

Maybe they meant that due to Russell's paradox there cannot be a 'universal set': a set of 'everything'.
Here's why: Assuming there would be a set of everything $U$, then we can consider the set $D$ defined as $\{ x \in U | x \not \in x \}$.  We could thus say that $D$ contains all 'normal' sets, where a 'normal' set is a set that does not contain itself as an element.
Now, since $U$ is universal, we have $D \in U$. But is $D \in D$? Well, if $D \in D$, then by definition of $D$, we don't put $D$ in $D$, i.e. $D \not \in D$.  But if $D \not\in D$, then by definition of $D$ we would put $D$ in $D$, i.e. $D \in D$. Hence, we obtain $D \in D$ if and only if $D \not \in D$, which is a contradiciton.
So, using the logic behind Russell's paradox, there cannot be a universal set. ... And yes, that would certainly be a set that can be said to be 'too big'.

