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Jumping through the transfinite: the master code hierarchy of Turing degrees1

  • Harold T. Hodes (a1)

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.

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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.

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[2] R. Boyd , G. Hensel and H. Putnam , A recursion-theoretic characterization of the ramified analytical hierarchy, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 4762.

[6] C. Jockusch and S. Simpson , A degree-theoretic characterization of the ramified analytical hierarchy, Annals of Mathematical Logic, vol. 10 (1976).

[7] S. Leeds and H. Putnam , An intrinsic characterization of the hierarchy of the constructible sets of integers, Logic Colloquium '69, North-Holland, Amsterdam and London, 1971.

[8] W. Marak and M. Srebeny , Gaps in the constructible universe, Annals of Mathematical Logic, vol. 6 (1974), pp. 359394.

[9] G. Sacks , Forcing with perfect closed sets, Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, R. I., 1971.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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