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

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
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2178/jsl/1102022214
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 53,682
Through your library

References found in this work BETA

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.
Foundations of Recursive Model Theory.Terrence S. Millar - 1978 - Annals of Pure and Applied Logic 13 (1):45.
A New Spectrum of Recursive Models.André Nies - 1999 - Notre Dame Journal of Formal Logic 40 (3):307-314.

Add more references

Citations of this work BETA

Η-Representation of Sets and Degrees.Kenneth Harris - 2008 - Journal of Symbolic Logic 73 (4):1097-1121.
Degree Spectra of Prime Models.Barbara F. Csima - 2004 - Journal of Symbolic Logic 69 (2):430-442.

Add more citations

Similar books and articles

Degree Spectra of Prime Models.Barbara F. Csima - 2004 - Journal of Symbolic Logic 69 (2):430 - 442.
Prime Numbers and Factorization in IE1 and Weaker Systems.Stuart T. Smith - 1992 - Journal of Symbolic Logic 57 (3):1057 - 1085.
Prime Matter and Actuality.Christopher Byrne - 1995 - Journal of the History of Philosophy 33 (2):197-224.
Prime Matter and Extension in Aristotle.Paul Studtmann - 2006 - Journal of Philosophical Research 31:171-184.
On Uniqueness of Prime Models.Saharon Shelah - 1979 - Journal of Symbolic Logic 44 (2):215-220.

Analytics

Added to PP index
2010-08-24

Total views
67 ( #142,052 of 2,349,382 )

Recent downloads (6 months)
19 ( #35,784 of 2,349,382 )

How can I increase my downloads?

Downloads

My notes