Cornell Universrty, Ithaca, New York 14853
Where a is a Turing degree and ξ is an ordinal < (ℵ1) L1 , the result of performing ξ jumps on a, a (ξ), is defined set-theoretically, using Jensen's fine-structure results. This operation appears to be the natural extension through (ℵ1) L1 of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operation.
(Received September 08 1977)
* Thanks to the referee for finding several major and many minor errors. Special thanks to F. Abramson for suggesting the use of modified Steel conditions in the proofs of Lemmas 1 and 2 under Case 3. Writing of this paper was in part supported by a Fellowship from the Mellon Foundation.