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By J.E. Fenstad, P. G. Hinman

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Hinman, Recursion in the Superjump, this volume. O. Gandy, General recursive functionals of finite type and hierarchies of functions (A paper given at the Symposium o n Mathematical Logic held a t the University of Clermont Ferrand, June 1962). J. Grilliot, Selection functions for recursive functionals, Notre Dame Journal of Formal Logic 10 (1969) 225-234. 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.

Thus 17e(4)l2 17e01. and so I{eo)(n)l Now to finish case 3. Notice that by the claim f ( n l ) €92 and I f(nl)l 2 min( lrnol, Inl\). Letxi denote the possibility imaginary object L. HARRINGTON 48 {el l H n 1 ( i )If. n E % then the xi’s have substance, and the claim implies that f(xi)E%p andf(xi) 2 m i n ( i m o I , IxiI). H. is not violated. Let p(rno,ml)= 3M 5e. As always Ip(mo,m1)I < P . So 13m * S e l 2 13n1*5e1(. It is now possible to show that 172 1 is F-recursively Mahlo. The proof will only be sketched as it is precisely the same as Richter’s argument in [Ri].

But Inl< lf(4)I+l{e(q))(n)I< 17e(4)l

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