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A Jump Operator in Set Recursion

✍ Scribed by Dag Normann

Publisher: John Wiley and Sons
Year: 1979
Tongue: English
Weight: 810 KB
Volume: 25
Category: Article
ISSN: 0044-3050
DOI: 10.1002/malq.19790251308

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✦ Synopsis

A JUMP OPERATOR IN SET RECURSION by DAG NORMA" in Oslo (Norway)

' ~+ ~# ( e , F ) =

k f 3 S is not a normal functional. Recursion in ''+3S does not satisfy stage comparison and that a subset of I is recursive in h+3S if and only if both it and its complement are semirecursive. The reason for this misbehaviour seems to be that h+3S(e, F) is defined only for total F, while we need information only from a part of F to compute k+3S(e, F ) .

I n E-recursion there are two natural candidates for the jump, either a complete Z*-definable set, or the spectrum itself. In defining S-recursion we choose the latter.

Definition. Define S-recursion from E-recursion by adding the following sclienie:

(2, j , z ) is total on its spectrum over z, e = ( 8 , c l , n>.

G is an application of R, we must check if this application actually is an application of Fel. We may assume as an induction-hypothesis that all applications of R in subcom-

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