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PERMUTATIONS OF THE INTEGERS INDUCE ONLY THE TRIVIAL AUTOMORPHISM OF THE TURING DEGREES

Published online by Cambridge University Press:  07 August 2018

BJØRN KJOS-HANSSEN
BJØRN KJOS-HANSSEN*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF HAWAI‘I AT MĀNOA HONOLULU, HI, USAE-mail:bjoern.kjos-hanssen@hawaii.edu
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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.

Keywords

Primary 03D28Secondary 03D32Turing degreesgroup of automorphismsorder theorypermutation groupcomputability
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 24 , Issue 2 , June 2018 , pp. 165 - 174
Copyright
Copyright © The Association for Symbolic Logic 2018 

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