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Spectra of theories and structures


Authors: Uri Andrews and Joseph S. Miller
Journal: Proc. Amer. Math. Soc. 143 (2015), 1283-1298
MSC (2010): Primary 03C57, 03D45
Published electronically: October 16, 2014
MathSciNet review: 3293742
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Abstract: We introduce the notion of a degree spectrum of a complete theory to be the set of Turing degrees that contain a copy of some model of the theory. We generate examples showing that not all degree spectra of theories are degree spectra of structures and vice versa. To this end, we give a new necessary condition on the degree spectrum of a structure, specifically showing that the set of PA degrees and the upward closure of the set of 1-random degrees are not degree spectra of structures but are degree spectra of theories.


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Additional Information

Uri Andrews
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
Email: andrews@math.wisc.edu

Joseph S. Miller
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
Email: jmiller@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12283-0
Received by editor(s): August 13, 2012
Received by editor(s) in revised form: June 6, 2013
Published electronically: October 16, 2014
Additional Notes: The second author was supported by the National Science Foundation under grant DMS-1001847.
Communicated by: Julia Knight
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.