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Prime models of computably enumerable degree

Published online by Cambridge University Press:  12 March 2014

Rachel Epstein
Rachel Epstein*
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA, E-mail: rachel@math.uchicago.edu, URL: http://www.math.uchicago.edu/~rachel/
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Abstract

We examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b′ = 0′), This extends Csima's result that every CAD theory has a low prime model. We also prove a density result for c.e. degrees of prime models. In particular, if c and d are c.e. degrees with d < c and c not low2 (c″ > 0″), then for any CAD theory T, there exists a c.e. degree b with d < b < c such that T has a prime model of degree b, where b can be chosen so that b′ is any degree c.e. in c with d′ ≤ b′. As a corollary, we show that for any degree c with 0 < c < 0′, every CAD theory has a prime model of low c.e. degree incomparable with c. We show also that every CAD theory has prime models of low c.e. degree that form a minimal pair, extending another result of Csima. We then discuss how these results apply to homogeneous models.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1373 - 1388
Copyright
Copyright © Association for Symbolic Logic 2008

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References

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