A Hierarchy Based on a Type Two Object

The object of hierarchy theory is to break down certain classes of predicates into subclasses of gradually increasing complexity. An excellent example is furnished by the hyperarithmetical hierarchy. Here each class in the hierarchy is obtained from the previous class by adding a single number quantifier. The Souslin-Kleene theorem [2, XXIV] then tells us that the predicates occurring in the hierarchy are just the A{ predicates. A direct generalization of this result to the next higher type presents difficulties. In fact, it appears from [6] that a hierarchy in which a single type one quantifier is added at each stage could not exhaust the Af predicates. However, results on higher type objects suggest a different generalization. Kleene [3, XLVIII] has shown that the A\ predicates are just the predicates recursive in the type two object E defined by