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Coding true arithmetic in the Medvedev and Muchnik degrees

Published online by Cambridge University Press:  12 March 2014

Paul Shafer
Paul Shafer*
Affiliation:
Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY 14853, USA, E-mail: pshafer@math.cornell.edu URL: http://www.math.cornell.edu/~pshafer/
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Abstract

We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order theory of true arithmetic are pairwise recursively isomorphic (obtained independently by Lewis, Nies, and Sorbi [7]). We then restrict our attention to the degrees of closed sets and prove that the following theories are pairwise recursively isomorphic: the first-order theory of the closed Medvedev degrees, the first-order theory of the compact Medvedev degrees, the first-order theory of the closed Muchnik degrees, the first-order theory of the compact Muchnik degrees, and the second-order theory of true arithmetic. Our coding methods also prove that neither the closed Medvedev degrees nor the compact Medvedev degrees are elementarily equivalent to either the closed Muchnik degrees or the compact Muchnik degrees.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 1 , March 2011 , pp. 267 - 288
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[1]Bianchini, Caterina and Sorbi, Andrea, A note on closed degrees of difficulty of the Medvedev lattice, Mathematical Logic Quarterly, vol. 42 (1996), no. 1, pp. 127–133.CrossRefGoogle Scholar
[2]Dyment, Elena Z., Certain properties of the Medvedev lattice, Mathematics of the USSR Sbornik, vol. 30 (1976), pp. 321–340.CrossRefGoogle Scholar
[3]Dyment, Elena Z., Exact bounds ofdenumerable collections of degrees of difficulty, Matematicheskie Zametki, vol. 28 (1980), no. 6, pp. 899–910.Google Scholar
[4]Hodges, Wilfrid, A shorter model theory, Cambridge University Press, 1997.Google Scholar
[5]Jankov, V. A., The calculus of the weak law of excluded middle, Mathematics of the USSR-Izvestiya, vol. 2 (1968), no. 5, pp. 997–1004.CrossRefGoogle Scholar
[6]Lerman, Manuel, Degrees of unsolvability, Springer-Verlag, 1983.CrossRefGoogle Scholar
[7]Lewis, Andrew E. M., Nies, André, and Sorbi, Andrea, The first order theories of the Medvedev and Muchnik lattices, 5th Conference on Computability in Europe, CiE 2009 (Ambos-Spies, Klaus, Löwe, Benedikt, and Merkle, Wolfgang, editors), Lecture Notes in Computer Science, vol. 5635, Springer, 2009, pp. 324–331.Google Scholar
[8]Lewis, Andrew E. M., Shore, Richard A., and Sorbi, Andrea, Topological aspects of the Medvedev lattice, to appear.Google Scholar
[9]Medvedev, Yuri T., Degrees of difficulty of the mass problems, Doklady Akademii Nauk SSSR (NS), vol. 104, 1955, pp. 501–504.Google Scholar
[10]Medvedev, Yuri T., Finite problems, Doklady Akademii Nauk SSSR (NS), vol. 142, 1962, pp. 1015–1018.Google Scholar
[11]Muchnik, Albert A., On strongandweak reducibilities of algorithmic problems, Sibirskii Matematkheskii Zhurnal, vol. 4 (1963), pp. 1328–1341.Google Scholar
[12]Nies, André, Shore, Richard A., and Slaman, Theodore A., Interpretability and definability in the recursively enumerable degrees, Proceedings of the London Mathematical Society, vol. 77 (1998), no. 02, pp. 241–291.CrossRefGoogle Scholar
[13]Shafer, Paul, Characterizing the join-irreducible Medvedev degrees, Notre Dame Journal of Formal Logic, to appear.Google Scholar
[14]Shore, Richard A., The theory of the degrees below 0′, Journal of the London Mathematical Society, vol. 24 (1981), pp. 1–14.CrossRefGoogle Scholar
[15]Simpson, Stephen G., First-order theory of the degrees of recursive unsolvability, Annals of Mathematics, vol. 105 (1977), no. 1, pp. 121–139.CrossRefGoogle Scholar
[16]Simpson, Stephen G., Subsystems of second order arithmetic, Cambridge University Press, 2009.CrossRefGoogle Scholar
[17]Skvortsova, Elena Z., A faithful interpretation of the intuitionistic propositional calculus by means of an initial segment of the Medvedev lattice, Sibirskii Matematicheskii Zhurnal, vol. 29 (1988), no. 1, pp. 171–178.Google Scholar
[18]Soare, Robert I., Recursively enumerable sets and degrees, Springer-Verlag, 1987.CrossRefGoogle Scholar
[19]Sorbi, Andrea, Some remarks on the algebraic structure of the Medvedev lattice, this Journal, vol. 55 (1990), no. 2, pp. 831–853.Google Scholar
[20]Sorbi, Andrea, Embedding Brouwer algebras in the Medvedev lattice, Notre Dame Journal of Formal Logic, vol. 32 (1991), no. 2, pp. 266–275.CrossRefGoogle Scholar
[21]Sorbi, Andrea, Some quotient lattices of the Medvedev lattice, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, vol. 37 (1991), no. 9–12, pp. 167–182.CrossRefGoogle Scholar
[22]Sorbi, Andrea, The Medvedev lattice of degrees of difficulty, Computability, enumerability, unsolvability: Directions in recursion theory (Cooper, S. B., Slaman, T. A., and Wainer, S. S., editors), London Mathematical Society Lecture Notes, vol. 224, Cambridge University Press, 1996, pp. 289–312.CrossRefGoogle Scholar
[23]Sorbi, Andrea and Terwijn, Sebastiaan A., Intermediate logics and factors of the Medvedev lattice, Annals of Pure and Applied Logic, vol. 155 (2008), no. 2, pp. 69–85.CrossRefGoogle Scholar