As is well known, the General Set Comprehension Principle (any class is a set) leads to the Russell Paradox (the class $x \notin x$ cannot be a set). As a result, set theories must restrict the Comprehension Principle to avoid self-reference. For example, in the case of ZFC, this is done by enumerating a small list of "safe" comprehension schema such as Separation.
Does General Comprehension lead to other types of paradox than Russell's? Or is this basically the only thing that can go wrong?
Best Answer
Early attempts to repair Russell's paradox tried simple patches, like forbidding the predicate $x\notin x$. But there are infinite families of predicates that all cause essentially the same problem. For example, let $P(x)$ be the predicate $\lnot\exists y. x\in y \wedge y\in x $. Then there is no set of all $x$ such that $P(x)$ holds. I think there is one of these for any cyclic directed graph; the original Russell predicate $x\notin x$ corresponds to the graph with one vertex and one directed edge.