Bounding Prime Models

Journal of Symbolic Logic 69 (4):1117 - 1142 (2004)
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Abstract

A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model U of T decidable in X. It is easy to see that $X = 0\prime$ is prime bounding. Denisov claimed that every $X <_{T} 0\prime$ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets $X \leq_{T} 0\prime$ are exactly the sets which are not $low_2$ . Recall that X is $low_2$ if $X\prime\prime$ $\leq_{T} 0\prime$ . To prove that a $low_2$ set X is not prime bounding we use a $0\prime$ -computable listing of the array of sets { Y : Y $\leq_{T}$ X } to build a CAD theory T which diagonalizes against all potential X-decidable prime models U of T. To prove that any $non-low_{2}$ ; X is indeed prime bounding, we fix a function f $\leq_T$ X that is not dominated by a certain $0\prime$ -computable function that picks out generators of principal types. Given a CAD theory T. we use f to eventually find, for every formula $\varphi (\bar{x})$ consistent with T, a principal type which contains it, and hence to build an X-decidable prime model of T. We prove the prime bounding property equivalent to several other combinatorial properties, including some related to the limitwise monotonic functions which have been introduced elsewhere in computable model theory

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Citations of this work

Η-representation of sets and degrees.Kenneth Harris - 2008 - Journal of Symbolic Logic 73 (4):1097-1121.
Computability of Homogeneous Models.Karen Lange & Robert I. Soare - 2007 - Notre Dame Journal of Formal Logic 48 (1):143-170.
Degree spectra of prime models.Barbara F. Csima - 2004 - Journal of Symbolic Logic 69 (2):430-442.

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References found in this work

Computable Models of Theories with Few Models.Bakhadyr Khoussainov, Andre Nies & Richard A. Shore - 1997 - Notre Dame Journal of Formal Logic 38 (2):165-178.
A New Spectrum of Recursive Models.André Nies - 1999 - Notre Dame Journal of Formal Logic 40 (3):307-314.
Foundations of recursive model theory.Terrence S. Millar - 1978 - Annals of Mathematical Logic 13 (1):45.

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