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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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