Journal of Symbolic Logic

Coding true arithmetic in the Medvedev and Muchnik degrees

Paul Shafer

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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.

Article information

Source
J. Symbolic Logic, Volume 76, Issue 1 (2011), 267-288.

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1294171000

Digital Object Identifier
doi:10.2178/jsl/1294171000

Mathematical Reviews number (MathSciNet)
MR2791348

Zentralblatt MATH identifier
1222.03049

Citation

Shafer, Paul. Coding true arithmetic in the Medvedev and Muchnik degrees. J. Symbolic Logic 76 (2011), no. 1, 267--288. doi:10.2178/jsl/1294171000. https://projecteuclid.org/euclid.jsl/1294171000


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