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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Trivial, strongly minimal theories are model complete after naming constants


Authors: Sergey S. Goncharov, Valentina S. Harizanov, Michael C. Laskowski, Steffen Lempp and Charles F. D. McCoy
Journal: Proc. Amer. Math. Soc. 131 (2003), 3901-3912
MSC (2000): Primary 03C10; Secondary 03C35, 03C57
Published electronically: February 24, 2003
MathSciNet review: 1999939
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Abstract: We prove that if $\mathcal{M}$ is any model of a trivial, strongly minimal theory, then the elementary diagram $\operatorname{Th}(\mathcal{M}_M)$ is a model complete $\mathcal{L}_M$-theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are $\boldsymbol{0}''$-decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is $\Sigma^0_5$.


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Additional Information

Sergey S. Goncharov
Affiliation: Institute of Mathematics, Siberian Branch of the RAS, 630090 Novosibirsk, Russia
Email: gonchar@math.nsc.ru

Valentina S. Harizanov
Affiliation: Department of Mathematics, George Washington University, Washington, DC 20052
Email: harizanv@gwu.edu

Michael C. Laskowski
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: mcl@math.umd.edu

Steffen Lempp
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: lempp@math.wisc.edu

Charles F. D. McCoy
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: P.O. Box 668, Notre Dame, Indiana 46556
Email: mccoy@math.wisc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06951-X
PII: S 0002-9939(03)06951-X
Keywords: Strongly minimal, trivial geometry, uncountably categorical, model complete, computable model, spectrum of computable models
Received by editor(s): February 28, 2002
Received by editor(s) in revised form: June 25, 2002
Published electronically: February 24, 2003
Additional Notes: This research was partially supported by the NSF Binational Grant DMS-0075899
The first author’s research was also partially supported by the Russian Foundation for Basic Research grant 99-01-00485. The third author’s research was partially supported by NSF grant DMS-0071746. The fourth author’s research was partially supported by NSF grant DMS-9732526 and by the Vilas Foundation of the University of Wisconsin. The fifth author’s research was partially supported by an NSF VIGRE Fellowship
The fourth author would also like to thank numerous other model theorists with whom he had discussed this problem over the past few years
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2003 American Mathematical Society