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

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G. Hinman, Hierarchies of effective descriptive set theory, Trans. Amer. Math. 142 (August 1969) 111-140. C. Kleene, Recursive functionals and quantifiers of finite types I, Trans. Amer. Math. 91 (1959) 1-52. N. Moschovakis, Hyperanalytic predicates, Trans. Amer. Math. 129 (1967) 249-282. A. O. M. Yates (North-Holland, Amsterdam, 1971). 25 7 -27 1. W. O. M. Yates (North-Holland, Amsterdam, 1971) 273-288. R. Shoenfield, A hierarchy based on a type-two object, Trans. Amer. Math. SOC. 134 (October, 1968) 103-108.

Then there is a partial %,-recursive function cp such that for all n , cp(n)$iff InIEdomf,inwhichcasef(InI)= Icp(n)l. Theorem 3 can be proven by combining remark ( 2 ) of 5 1 with a few applications of Theorem 2. Theorems 1 and 3 should now make it clear that /3 isF-admissible. But this, together with Theorem 3 and Lemma 1 for the case /3 = 1% 1, should make it equally clear that 1% I is F-recursively Mahlo. So to summarize. Theorem 4 I % 1 = p c . The %-admissible ordinals are just the F-admissible ordinals which are less than The partial %-recursive functions are just the partial functions which are C , over M (F).

Then for any F,T , and a, 3 ' ( T ) and aT = a and mT = $; Q(T, F)= o but (- S'(T)or aT # a or mT + $1; 0, if @(T,F)= 0 and 1, if 2, if @ ( T , P ) = 1. 40 - P. G. 1 1. For any a and F (a) F is acceptable F,** is total ;and (b) ifFisacceptable then l - s c ( 2 E , F ) G Proof. 8. l-x(F,**). 12. S+(a, F ) = &(Pi*); hence, S+ is partial recursive in I%. c) Proof. Suppose first S+(a, = 0, so for some n , { a } ( F ) = n. Hence Fl*(T[o,n) = 0. 1 1 recursive in F:*. Hence &(Pi*) = 0. Conversely, if &(&’:*) = 0, say F i * ( T ) = 0, then @(T,F)= 0 and 3’(T), so T is a correct computation tree involving only F.