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Local Definitions in Degree Structures: The Turing Jump, Hyperdegrees and Beyond

Published online by Cambridge University Press:  15 January 2014

Richard A. Shore
Richard A. Shore*
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USAE-mail: [email protected], URL: http://www.math.cornell.edu/~shore/
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Abstract

There are Π5 formulas in the language of the Turing degrees, D, with ≤, ⋁ and ⋀, that define the relations x″ ≤ y″, x″ = y″ and so x ∈ L2(y) = {x ≥ y ∣ x″ = y″} in any jump ideal containing 0(ω). There are also Σ6 & Π6 and Π8 formulas that define the relations w = x″ and w = x′, respectively, in any such ideal I. In the language with just ≤ the quantifier complexity of each of these definitions increases by one. On the other hand, no Π2 or Σ2 formula in the language with just ≤ defines L2 or x ∈ L2(y). Our arguments and constructions are purely degree theoretic without any appeals to absoluteness considerations, set theoretic methods or coding of models of arithmetic. As a corollary, we see that every automorphism of I is fixed on every degree above 0″ and every relation on I that is invariant under double jump or joining with 0″ is definable over I if and only if it is definable in second order arithmetic with set quantification ranging over sets whose degrees are in I. Similar direct coding arguments show that every hyperjump ideal I is rigid and biinterpretable with second order arithmetic with set quantification ranging over sets with hyperdegrees in I. Analogous results hold for various coarser degree structures.

Type
Communications
Information
Bulletin of Symbolic Logic , Volume 13 , Issue 2 , June 2007 , pp. 226 - 239
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

Abraham, U. and Shore, R. A. [1986], The degrees of constructibility below a Cohen real, Journal of the London Mathematical Society. Third Series, vol. 53, pp. 193–208.Google Scholar
Cooper, S. B. [1990], The jump is definable in the structure of the degrees of unsolvability (research announcement), Bulletin of the American Mathematical Society. New Series, vol. 23, pp. 151–158.Google Scholar
Cooper, S. B. [1993], On a conjecture of Kleene and Post, Department of Pure Mathematics Preprint Series, no. 7, University of Leeds.Google Scholar
Cooper, S. B. [2001], On a conjecture of Kleene and Post, Mathematical Logic Quarterly, vol. 47, pp. 3–33.3.0.CO;2-A>CrossRefGoogle Scholar
Downey, R. G., Jockusch, C.G. Jr., and Stob, M. [1990], Array nonrecursive sets and multiple permitting arguments, Recursion theory week (Ambos-Spies, K., Müller, G. H., and Sacks, G. E., editors), Springer-Verlag, Berlin, pp. 141–174.Google Scholar
Feferman, S. [1965], Some applications of the notion of forcing and generic sets, Fundamenta Mathematicae, vol. 56, pp. 325–345.Google Scholar
Harrington, L. and Kechris, A. S. [1975], A basis result for sets of reals with an application to minimal covers, Proceedings of the American Mathematical Society, vol. 53, pp. 445–448.Google Scholar
Harrington, L. and Shore, R. A. [1981], Definable degrees and automorphisms of D, Bulletin of the American Mathematical Society. New Series, vol. 4, pp. 97–100.Google Scholar
Jockusch, C. G. Jr. [2002], review of Cooper [2001], Math. Rev. MR1808943 (2002m: 03061).Google Scholar
Jockusch, C. G. Jr. and Shore, R. A. [1984], Pseudo-jump operators II: transfinite iterations, hierarchies and minimal covers, The Journal of Symbolic Logic, vol. 49, pp. 1205–1236.CrossRefGoogle Scholar
Jockusch, C. G. Jr. and Simpson, S. G. [1976], A degree theoretic definition of the ramified analytic hierarchy, Annals of Mathematical Logic, vol. 10, pp. 1–32.Google Scholar
Jockusch, C. G. Jr. and Soare, R. I. [1970], Minimal covers and arithmetical sets, Proceedings of the American Mathematical Society, vol. 25, pp. 856–859.CrossRefGoogle Scholar
Jockusch, C. G. Jr. and Solovay, R. M. [1977], Fixed points of jump preserving automorphisms of degrees, Israel Journal of Mathematics, vol. 26, pp. 91–94.Google Scholar
Kleene, S. C. and Post, E. L. [1954], The upper semi-lattice of degrees of recursive unsolvability, Annals of Mathematics. Second Series, vol. 59, pp. 379–407.Google Scholar
Lerman, M. and Shore, R. A. [1988], Decidability and invariant classes for degree structures, Transactions of the American Mathematical Society, vol. 310, pp. 669–692.Google Scholar
Sacks, G. E. [1990], Higher recursion theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin.Google Scholar
Selman, A. L. [1972], Applications of forcing to the degree theory of the arithmetic hierarchy, Proceedings of the London Mathematical Society, vol. 25, pp. 586–602.Google Scholar
Shore, R. A. [1981], The theory of the degrees below 0′, Journal of the London Mathematical Society, (3), vol. 24, pp. 1–14.Google Scholar
Shore, R. A. [1982], Finitely generated codings and the degrees r.e. in a degree d, Proceedings of the American Mathematical Society, vol. 84, pp. 256–263.Google Scholar
Shore, R. A. and Slaman, T. A. [1999], Defining the Turing jump, Mathematics Research Letters, vol. 6, pp. 711–722.Google Scholar
Shore, R. A. and Slaman, T. A. [2001a], A splitting theorem for n-REA degrees, Proceedings of the American Mathematical Society, vol. 129, pp. 3721–3728.Google Scholar
Shore, R. A. and Slaman, T. A. [2001b], A discrete splitting theorem for n-REA degrees, http://www.math.cornell.edu/~shore/papers/pdf/dsplit5.pdf.Google Scholar
Simpson, S. G. [1977], First order theory of the degrees of recursive unsolvability, Annals of Mathematics. Second Series, vol. 105, pp. 121–139.Google Scholar
Slaman, T. A. [1991], Degree structures, Proceedings of the international congress on mathematics, Kyoto 1990, Springer-Verlag, Tokyo, pp. 303–316.Google Scholar
Slaman, T. A. and Woodin, H. W. [1986], Definability in the Turing degrees, Illinois Journal of Mathematics, vol. 30, pp. 320–334.Google Scholar
Slaman, T. A. and Woodin, H. W. [2007], Definability in degree structures, Journal of Mathematical Logic, to appear.Google Scholar