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On the jump of an $ \alpha $-recursively enumerable set


Author: Richard A. Shore
Journal: Trans. Amer. Math. Soc. 217 (1976), 351-363
MSC: Primary 02F27
DOI: https://doi.org/10.1090/S0002-9947-1976-0424544-2
MathSciNet review: 0424544
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Abstract: We discuss the proper definition of the jump operator in $ \alpha $-recursion theory and prove a sample theorem: There is an incomplete $ \alpha $-r.e. set with jump $ 0''$ unless there is precisely one nonhyperregular $ \alpha $-r.e. degree. Thus we have a theorem in the first order language of Turing degrees with the jump which fails to generalize to all admissible $ \alpha $.


References [Enhancements On Off] (What's this?)

  • [1] R. B. Jensen, The fine structure of the constructible hierarchy. With a section by Jack Silver, Ann. Math. Logic 4 (1972), 229-308; erratum, 443. MR 46 # 8834. MR 0309729 (46:8834)
  • [2] R. Jhu, Contributions to axiomatic recursion theory and related aspects of $ \alpha $-recursion theory, Ph. D. Thesis, University of Toronto, 1973.
  • [3] G. E. Sacks, Metarecursion theory, Sets, Models and Recursion Theory (Proc. Summer School Math. Logic and Tenth Logic Colloq., Leicester, 1965), North-Holland, Amsterdam, 1967, pp. 243-263. MR 40 #7109. MR 0253896 (40:7109)
  • [4] -, Post's problem, admissible ordinals and regularity, Trans. Amer. Math. Soc. 124 (1966), 1-23. MR 34 #1183. MR 0201299 (34:1183)
  • [5] -, Higher recursion theory, Springer-Verlag (to appear). MR 1080970 (92a:03062)
  • [6] J. R. Shoenfield, Degrees of unsolvability, North-Holland, Amsterdam, 1971. MR 0340011 (49:4768)
  • [7] R. A. Shore, Splitting an $ \alpha $-recursively enumerable set, Trans. Amer. Math. Soc. 204 (1975), 65-77. MR 0379154 (52:60)
  • [8] -, The recursively enumerable $ \alpha $-degrees are dense, Ann. Math. Logic (to appear).
  • [9] -, The irregular and non-hyperregular $ \alpha $-r.e. degrees, Israel J. Math. (to appear).
  • [10] S. G. Simpson, Admissible ordinals and recursion theory, Ph. D. Thesis, M.I.T., 1971.
  • [11] -, Degree theory on admissible ordinals, Generalized Recursion Theory (Proc. of the Oslo Symposium), J. E. Fenstad and P. G. Hinman (Editors), North-Holland, Amsterdam, 1974, pp. 165-194. MR 0537203 (58:27405)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0424544-2
Keywords: $ \alpha $-recursion theory, admissible ordinals, $ \alpha $-recursively enumerable, $ \alpha $-degree, $ \alpha $-jump, priority arguments
Article copyright: © Copyright 1976 American Mathematical Society

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