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TURING DEGREE SPECTRA OF DIFFERENTIALLY CLOSED FIELDS

Published online by Cambridge University Press:  21 March 2017

DAVID MARKER  and
RUSSELL MILLER
DAVID MARKER
Affiliation:
DEPARTMENT OF MATHEMATICS STATISTICS, & COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT CHICAGO CHICAGO, IL, USA E-mail: marker@math.uic.edu
RUSSELL MILLER
Affiliation:
DEPARTMENT OF MATHEMATICS QUEENS COLLEGE – CUNY 65-30 KISSENA BLVD. QUEENS, NY 11367, USA and PH.D. PROGRAMS IN MATHEMATICS AND COMPUTER SCIENCE CUNY GRADUATE CENTER 365 FIFTH AVENUE NEW YORK, NY 10016, USA E-mail: russell.miller@qc.cuny.edu URL: http://qcpages.qc.cuny.edu/∼rmiller/
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Abstract

The degree spectrum of a countable structure is the set of all Turing degrees of presentations of that structure. We show that every nonlow Turing degree lies in the spectrum of some differentially closed field (of characteristic 0, with a single derivation) whose spectrum does not contain the computable degree 0. Indeed, this is an equivalence, for we also show that if this spectrum contained a low degree, then it would contain the degree 0. From these results we conclude that the spectra of differentially closed fields of characteristic 0 are exactly the jump-preimages of spectra of automorphically nontrivial graphs.

Keywords

Primary 03D45Secondary 03C5712H0503C10computabilitycomputable model theorydifferentially closed fieldsmodel theoryTuring degree spectrum

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 1 - 25
Copyright
Copyright © The Association for Symbolic Logic 2017 

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