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Degree spectra of prime models

Published online by Cambridge University Press:  12 March 2014

Barbara F. Csima
Barbara F. Csima*
Affiliation:
Department of Mathematics, Cornell University, New York 14853-4201, USA, E-mail: csima@math.cornell.edu
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Abstract.

We consider the Turing degrees of prime models of complete decidable theories. In particular we show that every complete decidable atomic theory has a prime model whose elementary diagram is low. We combine the construction used in the proof with other constructions to show that complete decidable atomic theories have low prime models with added properties.

If we have a complete decidable atomic theory with all types of the theory computable, we show that for every degree d with 0 < d < 0', there is a prime model with elementary diagram of degree d. Indeed, this is a corollary of the fact that if T is a complete decidable theory and L is a computable set of c.e. partial types of T, then for any degree d > 0, T has a d-decidable model omitting the nonprincipal types listed by L.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 2 , June 2004 , pp. 430 - 442
Copyright
Copyright © Association for Symbolic Logic 2004

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