Generic muchnik reducibility and presentations of fields

Abstract

We prove that if I is a countable ideal in the Turing degrees, then the field R _I of real numbers in I is computable from exactly the degrees that list the functions (i.e., elements of ω ^ω) in I. This implies, for example, that the degree spectrum of the field of computable real numbers consists exactly of the high degrees. We also prove that if I is a countable Scott ideal, then it is strictly easier to list the sets (i.e., elements of 2^ω) in I than it is to list the functions in I. This allows us to answer a question of Knight, Montalbán, and Schweber. They introduced generic Muchnik reducibility to extend the idea of Muchnik reducibility between countable structures to arbitrary structures. They asked if R is generically Muchnik reducible to the structure that consists of all sets of natural numbers. Our result for Scott ideals shows that this is not the case.

We finish by considering generic Muchnik reducibility of a countable structure A to an arbitrary structure B. We relate this to a couple of conditions asserting the ubiquity of countable elementary substructures of B that are Muchnik above A; we prove that one of these conditions is strictly stronger and the other is strictly weaker than generic Muchnik reducibility.