Trivial, strongly minimal theories are model complete after naming constants

Goncharov, Sergey; Harizanov, Valentina; Laskowski, Michael; Lempp, Steffen; McCoy, Charles

doi:10.1090/S0002-9939-03-06951-X

Trivial, strongly minimal theories are model complete after naming constants
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by Sergey S. Goncharov, Valentina S. Harizanov, Michael C. Laskowski, Steffen Lempp and Charles F. D. McCoy PDF

Proc. Amer. Math. Soc. 131 (2003), 3901-3912 Request permission

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 $\Sigma ^0_5$.

References

J. T. Baldwin and A. H. Lachlan, On strongly minimal sets, J. Symbolic Logic 36 (1971), 79–96. MR 286642, DOI 10.2307/2271517
Steven Buechler, Essential stability theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1996. MR 1416106, DOI 10.1007/978-3-642-80177-8
C. C. Chang and H. J. Keisler, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990. MR 1059055
Yuri L. Ershov and Sergei S. Goncharov, Constructive models, Siberian School of Algebra and Logic, Consultants Bureau, New York, 2000. MR 1749622, DOI 10.1007/978-1-4615-4305-3
S. S. Gončarov, Constructive models of $\aleph _{1}$-categorical theories, Mat. Zametki 23 (1978), no. 6, 885–888 (Russian). MR 502056
Goncharov, Sergei S. and Khoussainov, Bakhadyr M. On the complexity of theories of computable $\aleph _1$-categorical models, Vestnik NGU, Ser. Mat.-Mekh.-Inform. 1 (2) (2001), 63–76.
Goncharov, Sergei S. and Khoussainov, Bakhadyr M. Complexity of theories of computable models, Algebra and Logic, to appear.
Valentina S. Harizanov, Pure computable model theory, Handbook of recursive mathematics, Vol. 1, Stud. Logic Found. Math., vol. 138, North-Holland, Amsterdam, 1998, pp. 3–114. MR 1673621, DOI 10.1016/S0049-237X(98)80002-5
Leo Harrington, Recursively presentable prime models, J. Symbolic Logic 39 (1974), 305–309. MR 351804, DOI 10.2307/2272643
Bernhard Herwig, Steffen Lempp, and Martin Ziegler, Constructive models of uncountably categorical theories, Proc. Amer. Math. Soc. 127 (1999), no. 12, 3711–3719. MR 1610909, DOI 10.1090/S0002-9939-99-04920-5
N. G. Hisamiev, Strongly constructive models of a decidable theory, Izv. Akad. Nauk Kazah. SSR Ser. Fiz.-Mat. 1 (1974), 83–84, 94 (Russian). MR 0354344
Bakhadyr Khoussainov, Andre Nies, and Richard A. Shore, Computable models of theories with few models, Notre Dame J. Formal Logic 38 (1997), no. 2, 165–178. MR 1489408, DOI 10.1305/ndjfl/1039724885
K. Ž. Kudaĭbergenov, Constructivizable models of undecidable theories, Sibirsk. Mat. Zh. 21 (1980), no. 5, 155–158, 192 (Russian). MR 592228
Kueker, David W., Weak invariance and model completeness relative to parameters, in preparation.
David Marker, Non $\Sigma _n$ axiomatizable almost strongly minimal theories, J. Symbolic Logic 54 (1989), no. 3, 921–927. MR 1011179, DOI 10.2307/2274752
André Nies, A new spectrum of recursive models, Notre Dame J. Formal Logic 40 (1999), no. 3, 307–314. MR 1845630, DOI 10.1305/ndjfl/1022615611
Anand Pillay, An introduction to stability theory, Oxford Logic Guides, vol. 8, The Clarendon Press, Oxford University Press, New York, 1983. MR 719195
S. Shelah, Classification theory and the number of nonisomorphic models, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551