Proceedings of the American Mathematical Society

Author(s): Sergey S. Goncharov; Valentina S. Harizanov; Michael C. Laskowski; Steffen Lempp; Charles F. D. McCoy
Journal: Proc. Amer. Math. Soc. 131 (2003), 3901-3912.
MSC (2000): Primary 03C10; Secondary 03C35, 03C57
Posted: February 24, 2003
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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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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: 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
Posted: 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.
Copyright of article: Copyright 2003, American Mathematical Society

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