 Russell himself then tried to reduce mathematics to logic in another way. Frege’s Basic Law V entails that corresponding to every property of mathematical entities, there exists a class of mathematical entities having that property. This was evidently too strong, for it was exactly this consequence which led to Russell’s paradox. So Russell postulated that only properties of mathematical objects that have already been shown to exist, determine classes. Predicates that implicitly refer to the class that they were to determine if such a class existed, do not determine a class. Thus a typed structure of properties is obtained: properties of ground objects, properties of ground objects and classes of ground objects, and so on. This typed structure of properties determines a layered universe of mathematical objects, starting from ground objects, proceeding to classes of ground objects, then to classes of ground objects and classes of ground objects, and so on.