## The $\Delta _3^0$-automorphism method and noninvariant classes of degrees

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- by Leo Harrington and Robert I. Soare PDF
- J. Amer. Math. Soc.
**9**(1996), 617-666 Request permission

## Abstract:

A set $A$ of nonnegative integers is*computably enumerable (c.e.)*, also called

*recursively enumerable (r.e.)*, if there is a computable method to list its elements. Let $\mathcal {E}$ denote the structure of the computably enumerable sets under inclusion, $\mathcal {E} = ( \{ W_e \}_{e\in \omega }, \subseteq )$. Most previously known automorphisms $\Phi$ of the structure $\mathcal {E}$ of sets were effective (computable) in the sense that $\Phi$ has an effective presentation. We introduce here a new method for generating noneffective automorphisms whose presentation is $\Delta ^0_3$, and we apply the method to answer a number of long open questions about the orbits of c.e. sets under automorphisms of $\mathcal {E}$. For example, we show that the orbit of every noncomputable (

*i.e.*, nonrecursive) c.e. set contains a set of high degree, and hence that for all $n>0$ the well-known degree classes $\mathbf {L}_n$ (the low$_n$ c.e. degrees) and $\overline {\mathbf {H}}_n = \mathbf {R} - \mathbf {H}_n$ (the complement of the high$_n$ c.e. degrees) are noninvariant classes.

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## Additional Information

**Leo Harrington**- Affiliation: Department of Mathematics University of California at Berkeley Berkeley, California 94720
- Email: leo@math.berkeley.edu
**Robert I. Soare**- Affiliation: Department of Mathematics University of Chicago 5734 University Avenue Chicago, Illinois 60637-1538;
*World Wide Web address*: http://www.cs.uchicago.edu/~soare - Email: soare@math.uchicago.edu
- Received by editor(s): December 13, 1993
- Additional Notes: The first author was supported by National Science Foundation Grants DMS 89-10312 and DMS 92-14048, and the second author by National Science Foundation Grants DMS 91-06714 and DMS 95-00825
- © Copyright 1996 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**9**(1996), 617-666 - MSC (1991): Primary 03D25
- DOI: https://doi.org/10.1090/S0894-0347-96-00181-6
- MathSciNet review: 1311821