Permutations of the Integers Induce Only the Trivial Automorphism of the Turing Degrees

Abstract

Is there a nontrivial automorphism of the Turing degrees? It is a major open problem of computability theory. Past results have limited how nontrivial automorphisms could possibly be. Here we consider instead how an automorphism might be induced by a function on reals, or even by a function on integers. We show that a permutation of ω cannot induce any nontrivial automorphism of the Turing degrees of members of 2ω, and in fact any permutation that induces the trivial automorphism must be computable.A main idea of the proof is to consider the members of 2ω to be probabilities, and use statistics: from random outcomes from a distribution we can compute that distribution, but not much more.