TITLE: How is Cantor's diagonal argument related to Russell's paradox in naive set theory?
QUESTION [9 upvotes]: I was wondering whether anyone can shed proper light on this issue. I read both and it seems like they are somewhat similar, yet I can't quite see it.

REPLY [7 votes]: It is perhaps worth recalling that in his Introduction to Mathematical Philosophy, Russell writes

When I first came upon this contradiction [in the idea that there is a
  greatest cardinal], in the year 1901, I attempted to discover some flaw
  in Cantor's proof that there is no greatest cardinal ... Applying this
  proof to the supposed class of all imaginable objects, I was led to a
  new and simpler contradiction, namely, the following :-
The comprehensive class we are considering, which is to embrace
  everything, must embrace itself as one of its members. In other words,
  if there is such a thing as "everything" then "everything" is
  something, and is a member of the class " everything." But normally a
  class is not a member of itself. Mankind, for example, is not a man.
  Form now the assemblage of all classes which are not members of
  themselves. This is a class: is it a member of itself or not? If it
  is, it is one of those classes that are not members of themselves,
  i.e. it is not a member of itself. If it is not, it is not one of
  those classes that are not members of themselves, i.e. it is a member
  of itself. Thus of the two hypotheses - that it is, and that it is
  not, a member of itself -  each implies its contradictory. This is a
  contradiction.

So yes, Russell came upon his paradox in analysing what is happening in the Cantor proof applied to the limiting case of a (supposed) universal set. There is indeed that close relationship between the arguments.