By J.E. Fenstad and P.G. Hinman (Eds.)

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Extra resources for Generalized Recursion Theory: Proceedings of the 1972 Oslo Symposium

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C. Kleene, Recursive functionals and quantifiers of finite type, Trans. Am. Math. Soc. 91 (1959) 1-52; 108 (1963) 106-142. A. O. M. ) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) pp. 257271. B. D. dissertation, MIT (1 972). N. Moschovakis, Hyperanalytic predicates, Trans. Am. Math. Soc. 129 (1967) 249-282. [Ri] W. O. M. Yates (eds) Logic Colloquium '69 (North-Holland, Amsterdam, 1971) pp. 273-288. E. Sacks, The 1-Section of a type n object, this volume. R. Shoenfield, A hierarchy based on a type 2 object, Trans.

Math. Soc. 134 (1968) 103-108. Fenstad, P. J. Generalized Recursion Theory @ North-Holland Publ. , I974 STRUCTURAL CHARACTERIZATIONS OF CLASSES OF RELATIONS Yiannis N. MOSCHOVAKIS University of California, Los Angeles For each object U of finite type over the integers and for each k 2 1 , put ken(U) = the k-envelope of U = the set of all relations with arguments of type < k which are semirecursive in U . e. the existential quantifier mE over the objects of type m - 2 is recursive in U . g. the E and the (positive) second order inductively definable relations with arguments of type 0 and 1.

So is P defined by P(x) - (3ai)R(ai,x). Closure under V1,& and v is defined similarly. J is in r and f : 5X + y is primitive recursive, then f-' [ R ]= {x : f(x) E R } is also in r. Using the basic definitions and the Stage Comparison Theorem, one easily proves that i f U is normal of type m 2 2 and 1 <_ k 5 m + 1, then ken ( U )is closed under primitive recursive substitution, &, V, 3' and V j for every jlm-2. A k-pointclass r is o-parametrized if for every space X of type < k there is a relation G 5 o X 5X in r such that for every R E X, R isin forsome e E o , R = G, = {x E X : G ( e , x ) }.

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