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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Pseudojump operators. I. The r.e. case
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by Carl G. Jockusch and Richard A. Shore PDF
Trans. Amer. Math. Soc. 275 (1983), 599-609 Request permission

Abstract:

Call an operator $J$ on the power set of $\omega$ a pseudo jump operator if $J(A)$ is uniformly recursively enumerable in $A$ and $A$ is recursive in $J(A)$ for all subsets $A$ of $\omega$. Thus the (Turing) jump operator is a pseudo jump operator, and any existence proof in the theory of r.e. degrees yields, when relativized, one or more pseudo jump operators. Extending well-known results about the jump, we show that for any pseudo jump operator $J$, every degree $\geqslant {\mathbf {0}}’$ has a representative in the range of $J$, and that there is a nonrecursive r.e. set $A$ with $J(A)$ of degree ${\mathbf {0}}’$. The latter result yields a finite injury proof in two steps that there is an incomplete high r.e. degree, and by iteration analogous results for other levels of the ${H_n}$, ${L_n}$ hierarchy of r.e. degrees. We also establish a result on pairs of pseudo jump operators. This is combined with Lachlan’s result on the impossibility of combining splitting and density for r.e. degrees to yield a new proof of Harrington’s result that ${\mathbf {0}}’$ does not split over all lower r.e. degrees.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 275 (1983), 599-609
  • MSC: Primary 03D25; Secondary 03D30
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0682720-1
  • MathSciNet review: 682720