Journal of Symbolic Logic

Prime models of computably enumerable degree

Rachel Epstein

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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.

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J. Symbolic Logic, Volume 73, Issue 4 (2008), 1373-1388.

First available in Project Euclid: 27 December 2008

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Epstein, Rachel. Prime models of computably enumerable degree. J. Symbolic Logic 73 (2008), no. 4, 1373--1388. doi:10.2178/jsl/1230396926.

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