Sai Sasank Y (sai-sasank-y)
Naïve Set Theory - Part 1: Construction of Sets
From what I understand, such predicates seem to be causing trouble. For example, the result that no set contains everything seems like too strong a result at this point.
From the book: "To specify a set, it is not enough to pronounce some magic words; it is also necessary to have at hand a set to whose elements the magic words apply". Magic words basically mean the predicates S(x).
The book says such x's don't constitute a set and calls them illegal. It also mentions that class is the word to describe such x's and that classes are irrelevant in its approach to set theory.
Perhaps when I read further I'd be able to reason better.