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

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**Example text**

In the introduction we defined S as a function on jumps. 1 ; if (a(0))(a+, 1, otherwise. This form is more convenient for treating computations in S . 17. There exists a primitive recursive 71 such that for any total F and any x , i f F E 3 '2 with I 6 1 -index x , then S ( F ) E 9? (X>. Proof. Suppose F E W 2 with 161-index x . ;Q). 1 is uniformly semicompu table from J,cf(x,v)) and hence computable from S(J,cfcx,y)))which isJ2eu(x,y)). From this it is easy to compute a I 61-index for S(F). 18.

We shall continue to use F , G , H , ... to denote total type 2 objects while we shall use F,G, ... to denote partial type 2 objects. 2 introducing the new type of variable: with appropriate modifications in the indexing to take care of the new scheme and the new type of variable. , ql of functionals, some of whose arguments may be partial. ') = \ki(a). 8 of [KI]) would no longer be monotone, so that a crucial ingredient of [Kl] would be missing. , are consistent then cp is also consistent. It is important to distinguish between variables F and F even in some contexts where this might not appear necessary.

2E is a type 2 functional defined by 0 if X = @ 1 otherwise (where X is any real). i 43 44 L. HARRINGTON Let F be a fixed type 2 functional. F will stay more or less fixed till further notice. The rest of this paper will be developed relative to F. This will seemingly increase the generality of the results while in no way increasing the difficulty of their proofs. In fact F will in general be tacitly ignored. Definition. 1 - Sc (F,2E), [ 1 - Sc (F,S)], is the collection of reds recursive in F, 2~ [recursive in F, SI.