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

DOI:
https://doi.org/10.1090/S0002-9939-03-06951-X

Published electronically:
February 24, 2003

MathSciNet review:
1999939

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if is any model of a trivial, strongly minimal theory, then the elementary diagram is a model complete -theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are -decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is .

**1.**Baldwin, John T. and Lachlan, Alistair H.,*On strongly minimal sets*, J. Symbolic Logic**36**(1971), 79-96. MR**44:3851****2.**Buechler, Steven A.,*Essential Stability Theory*, Springer-Verlag, Heidelberg, 1996. MR**98j:03050****3.**Chang, C. C. and Keisler, H. Jerome,*Model Theory (3rd edition)*, North-Holland, Amsterdam, 1990. MR**91c:03026****4.**Ershov, Yuri L. and Goncharov, Sergei S.*Constructive models*, Siberian School of Algebra and Logic, Consultants Bureau, New York, 2000. MR**2002a:03069****5.**Goncharov, Sergei S.,*Constructive models of -categorical theories*, Mat. Zametki**23**(1978), 885-888. MR**80g:03029****6.**Goncharov, Sergei S. and Khoussainov, Bakhadyr M.*On the complexity of theories of computable -categorical models*, Vestnik NGU, Ser. Mat.-Mekh.-Inform.**1**(2) (2001), 63-76.**7.**Goncharov, Sergei S. and Khoussainov, Bakhadyr M.*Complexity of theories of computable models*, Algebra and Logic, to appear.**8.**Harizanov, Valentina S.,*Pure computable model theory*, in:*``Handbook of recursive mathematics''*, Vol. 1, North-Holland, Amsterdam, 1998, pp. 3-114. MR**2000f:03108****9.**Harrington, Leo,*Recursively presented prime models*, J. Symbolic Logic**39**(1974), 305-309. MR**50:4292****10.**Herwig, Bernhard; Lempp, Steffen; and Ziegler, Martin,*Constructive models of uncountably categorical theories*, Proc. Amer. Math. Soc.*127*(1999), 3711-3719. MR**2000b:03129****11.**Khisamiev, Nazif G.,*On strongly constructive models of decidable theories*, Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat.**35**(1) (1974), 83-84. MR**50:6824****12.**Khoussainov, Bakhadyr M.; Nies, André O.; and Shore, Richard A.,*Computable models of theories with few models*, Notre Dame J. Formal Logic**38**(1997), 165-178. MR**99c:03049****13.**Kudaibergenov, Kanat Zh.,*Constructivizable models of undecidable theories*, Sibirsk. Mat. Zh.**21**(5) (1980), 155-158, 192. MR**82h:03040****14.**Kueker, David W.,*Weak invariance and model completeness relative to parameters*, in preparation.**15.**Marker, David,*Non axiomatizable almost strongly minimal theories*, J. Symbolic Logic**54**(1989), 921-927. MR**90g:03037****16.**Nies, André O.,*A new spectrum of recursive models*, Notre Dame J. Formal Logic**40**(1999), 307-314. MR**2002e:03066****17.**Pillay, Anand,*An introduction to stability theory*, Oxford University Press, New York, 1983. MR**85i:03104****18.**Shelah, S.*Classification theory and the number of nonisomorphic models*, 2nd edition, North-Holland Publishing Co., Amsterdam, 1990. MR**91k:03085**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
03C10,
03C35,
03C57

Retrieve articles in all journals with MSC (2000): 03C10, 03C35, 03C57

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:
https://doi.org/10.1090/S0002-9939-03-06951-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