Recursive spectra of strongly minimal theories satisfying the Zilber Trichotomy

Andrews, Uri; Medvedev, Alice

doi:10.1090/S0002-9947-2014-05897-2

Recursive spectra of strongly minimal theories satisfying the Zilber Trichotomy
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Trans. Amer. Math. Soc. 366 (2014), 2393-2417 Request permission

Abstract:

We conjecture that for a strongly minimal theory $T$ in a finite signature satisfying the Zilber Trichotomy, there are only three possibilities for the recursive spectrum of $T$: all countable models of $T$ are recursively presentable; none of them are recursively presentable; or only the zero-dimensional model of $T$ is recursively presentable. We prove this conjecture for disintegrated (formerly, trivial) theories and for modular groups. The conjecture also holds via known results for fields. The conjecture remains open for finite covers of groups and fields.

References

Uri Andrews, A new spectrum of recursive models using an amalgamation construction, J. Symbolic Logic 76 (2011), no. 3, 883–896. MR 2849250, DOI 10.2178/jsl/1309952525
Uri Andrews, New spectra of strongly minimal theories in finite languages, Ann. Pure Appl. Logic 162 (2011), no. 5, 367–372. MR 2765579, DOI 10.1016/j.apal.2010.11.002
J. T. Baldwin and A. H. Lachlan, On strongly minimal sets, J. Symbolic Logic 36 (1971), 79–96. MR 286642, DOI 10.2307/2271517
Thomas Blossier and Elisabeth Bouscaren, Finitely axiomatizable strongly minimal groups, J. Symbolic Logic 75 (2010), no. 1, 25–50. MR 2605881, DOI 10.2178/jsl/1264433908
Steven Buechler, Essential stability theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1996. MR 1416106, DOI 10.1007/978-3-642-80177-8
S. S. Gončarov, Constructive models of $\aleph _{1}$-categorical theories, Mat. Zametki 23 (1978), no. 6, 885–888 (Russian). MR 502056
Sergey S. Goncharov, Valentina S. Harizanov, Michael C. Laskowski, Steffen Lempp, and Charles F. D. McCoy, Trivial, strongly minimal theories are model complete after naming constants, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3901–3912. MR 1999939, DOI 10.1090/S0002-9939-03-06951-X
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
Ehud Hrushovski, A new strongly minimal set, Ann. Pure Appl. Logic 62 (1993), no. 2, 147–166. Stability in model theory, III (Trento, 1991). MR 1226304, DOI 10.1016/0168-0072(93)90171-9
U. Hrushovski and A. Pillay, Weakly normal groups, Logic colloquium ’85 (Orsay, 1985) Stud. Logic Found. Math., vol. 122, North-Holland, Amsterdam, 1987, pp. 233–244. MR 895647, DOI 10.1016/S0049-237X(09)70556-7
Ehud Hrushovski and Boris Zilber, Zariski geometries, J. Amer. Math. Soc. 9 (1996), no. 1, 1–56. MR 1311822, DOI 10.1090/S0894-0347-96-00180-4
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
Daniel Lascar, Omega-stable groups, Model theory and algebraic geometry, Lecture Notes in Math., vol. 1696, Springer, Berlin, 1998, pp. 45–59. MR 1678598, DOI 10.1007/978-3-540-68521-0_{3}
Angus Macintyre, The word problem for division rings, J. Symbolic Logic 38 (1973), 428–436. MR 337554, DOI 10.2307/2273039
David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. MR 1924282
Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
Bruno Poizat, MM. Borel, Tits, Zil′ber et le Général Nonsense, J. Symbolic Logic 53 (1988), no. 1, 124–131 (French). MR 929379, DOI 10.2307/2274432