Skip to main content
×
Home
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 5
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Zhu, Yizheng 2015. Realizing an <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd"><mml:msup><mml:mrow><mml:mtext>AD</mml:mtext></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math> model as a derived model of a premouse. Annals of Pure and Applied Logic, Vol. 166, Issue. 12, p. 1275.


    Rudominer, Mitch 2000. Inner model operators in L(R). Annals of Pure and Applied Logic, Vol. 101, Issue. 2-3, p. 147.


    Shore, Richard A. 1999. Handbook of Computability Theory.


    Weber, Frank P. 1995. Invariant Constructions of Simple and Maximal Sets. Mathematical Logic Quarterly, Vol. 41, Issue. 2, p. 143.


    Welch, P.D. 1987. Logic Colloquium '86, Proceedings of the Colloquium held in Hull.


    ×

A classification of jump operators

  • John R. Steel (a1)
  • DOI: http://dx.doi.org/10.2307/2273146
  • Published online: 01 March 2014
Abstract

The structure (D, ≤) of the Turing degrees under Turing reducibility is quite complicated. This is true even if we restrict our attention to the substructure (R, ≤) of r.e. degrees. However, the theorems which imply that these structures are complicated all involve ad hoc constructions of sets having the desired reducibility relations. The complexity disappears when we turn to degrees occurring in nature. Of the degrees in R, only 0 and 0′ seem natural. In D, only 0, 0′, 0″, …, 0ω, 0ω+1, … (and on into the transfinite) seem natural. Thus the natural degrees appear to be wellordered, with successors given by Turing jump.

If this is true, one would like to prove it. Of course the first problem is to make the concept of naturalness more precise. The following requirements seem plausible: a natural degree should be definable, its definition should relativise to an arbitrary degree, and this relativisation should preserve reducibility relations among natural degrees. Thus to each natural degree c is associated a definable fc: DD so that fc(0) = c and ∀d(dfc(d)). Moreover, bc iff ∀d(fb(d) ≤, fc(d)). To be specific, let us take the definability of fc to mean that fcL(R).

If P is a property of degrees, we say P holds almost everywhere (a.e.) iff ∃cd ≥: c P(d). For f, g: DD, let fmg iff f(d) ≤ g(d) a.e. Define f′ by f′(d) = f{d)′, and let M = {f: DD/fL(R) ∧ df(d) a.e.}. The following conjecture is due to D. A. Martin:

Conjecture. M is prewellordered by ≤m. If fM has rank α in ≤m, then f′ has rank α + 1.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]H. Becker , Partially playful universes, Cabal Seminar, 1976–1977, Lecture Notes in Mathematics, vol. 689, Springer-Verlag, Berlin and New York, 1978, pp. 5591.

[3]R. B. Jensen , The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229308.

[4]C. Jockusch and S. Simpson , A degree-theoretic definition of the ramified analytic hierarchy, Annals of Mathematical Logic, vol. 10 (1976), pp. 132.

[5]A. S. Kechris , AD and projective ordinals, Cabal Seminar, 1976–1977, Lecture Notes in Mathematics, vol. 689, Springer-Verlag, Berlin and New York, 1978, pp. 91133.

[7]D. A. Martin , The axiom of determinateness and reduction principle in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687689.

[8]G. E. Sacks , On a theorem of Lachlan and Martin, Proceedings of the American Mathematical Society, vol. 18 (1967), pp. 140141.

[11]R. Van Wesep , Wadge degrees and descriptive set theory, Cabal Seminar, 1976–1977, Lecture Notes in Mathematics, vol. 689, Springer-Verlag, Berlin and New York, 1978, pp. 151171.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×