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

ISBN-10: 044410545X

ISBN-13: 9780444105455

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P. ), Generalized Recursion Theory @North-Holland Publ. The main result of this paper is that of, the first ordinal not recursive in the superjump, is the first recursively Mahlo ordinal. This result was announced in [Ac], but its proof hinged on some corollaries of a fallacious theorem from [Pl] . These corollaries turn out to be correct. As prerequisites, a working knowledge of Kleene’s (or some equivalent) formulation of recursion on higher type functionals, [Kl] ,would be desirable. The reader would also be advised to acquaint himself with the discussions in [Pl] and [A-HI, and most particularly with $ 3 and $4 of [A-HI.

F~ is F-recursively Mahlo if u is F-admissible and if for any function f : u + a, A, over M J F ) , there is an F-admissible /3 < u such that f”/3 Go. wf(p:) is the first F-admissible (F-recursively Mahlo) ordinal. In [Sh] , Shoenfield constructed a hierarchy for 1 - Sc(F, 2 E ) which could be used to show that 1 - Sc (F, 2 E )= 2w f’ L F ( F ) . This is done in a way totally analogous to the extension in [A-HI of the hierarchy from [Pl]. The technique of [Ri] is the guiding influence in both cases.

This follows from the fact that if fi is acceptable and F E G, then G is also acceptable, for any a and m,{ a } (m, 2E, k)= ( a } (m, 2E,G) (because only values k(a) for a recursive in 2E, F are used in the computation), and 1-sc( 2 E ,k)= 1-sc( 2 E , G). 3. (a) 8 is recursive in &+; (b) S is recursive in S + ; (c) & is recursive in S ; Proof. Hence P. G. HINMAN 36 w h e r e e E w such that { e ) ( a , ( 2 E , F ) ) =Oif 2E(ha{b}(a,2E,F))=0, and is undefined otherwise. O)(e). 0 Our aim now is to show that Z+ and S+ are both partial recursive in G .

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Generalized Recursion Theory: Proceedings of the 1972 Oslo Symposium by J.E. Fenstad and P.G. Hinman (Eds.)


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