81. Introduction
Iterations of the superjump operator have been studied by PLATEK [7], ACZEL-HIXMAN [I] and HARRIKGTON [4]. In this paper we treat the superjump operator as a map S from P("w), the power set of wm, into P("o). As such one quickly observes that the family of A; sets of reals is closed under X and certain transfinite iterations of 8. We show this closure extends to any iteration of X along a A: prewellordering of reals. 1 would like t,o thank STEVE SIMPSON for his guidance as I obtained these results for m y thesis [9].
8 2. Hltwit~ rcwrsion and t.he superjump Recursioii in higher types was first introduced by KLEENE [5] and the jump operator for that tlieory was discovered by GANDY [3]. Definition 2.1. Given L4, B 5 ow; A is Klee?ie semi-recursive in B iff (in the sense of KLEESE [ S ] ) there is a y E ("o and an e E uo so that A = {x: (el (r, y, Xu, 2 E ) j ] and . 4 i s Kileetie rec'ur. e. in. B (written ,4 sk B ) iff both A and "'cu -A are Kleene semire c11 rs i 1-e in B. Definition 2.2. Given B ww and y E "o let L , ( 3 ; y) = w LJ -(MI, and define
L,+,(R; y) to be the class of sets which are first order definable over the structure (Lp( R : y). E IL,(B; y). B A LB(B; y)) with parameters. We usually abbreviate this triple 1);i~ L,(B; y). If il is a limit, ordinal define LI(B; y) = I J LB(B; y). The least ordiiial for which L,(B; y) is admissible is denoted by m,(B; y). If / 3 = ojl(B; y), then L,(B; y) is an example of a ( B ; y)-admissible set.
and define (z),, = x and ( z ) ~ = y. If B ( ' c u let IzJB denote the least, ordinal p < co,(B; z ) so that (z),(O) is the Godel code for a Zl formula p(u: c) of set theory (in which there may occur a unary predicate symbol to be interpreted as B ) and L,+,(B; z ) k q((z),,, ( z ) ~) .
Note IzlIj may not always be defined. We denote the doniain of the map I by S ( B ) ; i.e. S(B) = ( z E Ow: I z ( B < N,>. The map S : P("'tu) -+ P("o) is called the superjump operator. p < 1.
We persistently rely on the observation that the following are equivalent : (i) , 4 is Kleene-semirecursive in B ; '17 %r.f.lir. f. math. Lugik