I think this article lacks some few words, I had to look up the wikipedia article to get wiser. The paradox is not about "the set of all sets", but "the set of all sets that are not members of themselves".
Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves.