Spectra of theories and structures

Andrews, Uri; Miller, Joseph

doi:10.1090/S0002-9939-2014-12283-0

Spectra of theories and structures
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by Uri Andrews and Joseph S. Miller

Proc. Amer. Math. Soc. 143 (2015), 1283-1298

DOI: https://doi.org/10.1090/S0002-9939-2014-12283-0

Published electronically: October 16, 2014

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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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