Abstract
In the 1960s, Clement F. Kent showed that there are continuum many permutations of \(\omega \) that map computable sets to computable sets. Thus these permutations preserve the bottom Turing degree 0. We show that a permutation of \(\omega \) cannot induce any nontrivial automorphism of the Turing degrees of members of \(2^{\omega }\), and in fact any permutation that induces the trivial automorphism must be computable.
This work was partially supported by a grant from the Simons Foundation (#315188 to Bjørn Kjos-Hanssen). This material is based upon work supported by the National Science Foundation under Grant No. 1545707. The author acknowledges the support of the Institut für Informatik at the University of Heidelberg, Germany during the workshop on Computability and Randomness, June 15 – July 9, 2015.
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Kjos-Hanssen, B. (2017). Permutations of the Integers Induce only the Trivial Automorphism of the Turing Degrees. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_35
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DOI: https://doi.org/10.1007/978-3-319-50062-1_35
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