December 2008 Prime models of computably enumerable degree
Rachel Epstein
J. Symbolic Logic 73(4): 1373-1388 (December 2008). DOI: 10.2178/jsl/1230396926


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.


Download Citation

Rachel Epstein. "Prime models of computably enumerable degree." J. Symbolic Logic 73 (4) 1373 - 1388, December 2008.


Published: December 2008
First available in Project Euclid: 27 December 2008

zbMATH: 1155.03017
MathSciNet: MR2467224
Digital Object Identifier: 10.2178/jsl/1230396926

Rights: Copyright © 2008 Association for Symbolic Logic


This article is only available to subscribers.
It is not available for individual sale.

Vol.73 • No. 4 • December 2008
Back to Top