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Coding true arithmetic in the Medvedev and Muchnik degrees. (English) Zbl 1222.03049

The author proves that the first-order theories of the Medvedev degrees and the Muchnik degrees are recursively isomorphic to the third-order theory of true arithmetic. Further, the author proves that the first-order theories of the closed Medvedev degrees and of the compact Medvedev degrees are recursively isomorphic to the second-order theory of true arithmetic. At last, it is proved 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.

MSC:

03D60 Computability and recursion theory on ordinals, admissible sets, etc.
03C57 Computable structure theory, computable model theory
03C62 Models of arithmetic and set theory
03C85 Second- and higher-order model theory
03D30 Other degrees and reducibilities in computability and recursion theory
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