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# A classification of jump operators

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.

References
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[1]Becker H., Partially playful universes, Cabal Seminar, 1976–1977, Lecture Notes in Mathematics, vol. 689, Springer-Verlag, Berlin and New York, 1978, pp. 5591.
[2]Hodes H., Jumping through the transfinite: the master-code hierarchy of Turing degrees, this Journal, vol. 45 (1980), pp. 204220.
[3]Jensen R. B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229308.
[4]Jockusch C. and Simpson S., A degree-theoretic definition of the ramified analytic hierarchy, Annals of Mathematical Logic, vol. 10 (1976), pp. 132.
[5]Kechris A. S., AD and projective ordinals, Cabal Seminar, 1976–1977, Lecture Notes in Mathematics, vol. 689, Springer-Verlag, Berlin and New York, 1978, pp. 91133.
[6]Lachlan A. H., Uniform enumeration operation, this Journal, vol. 40 (1975), pp. 401409.
[7]Martin D. A., The axiom of determinateness and reduction principle in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687689.
[8]Sacks G. E., On a theorem of Lachlan and Martin, Proceedings of the American Mathematical Society, vol. 18 (1967), pp. 140141.
[10]Steel J., Descending sequences of degrees, this Journal, vol. 40 (1975), pp. 5961.
[11]Wesep R. Van, 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.
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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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