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Finitely axiomatizable ω-categorical theories and the Mazoyer hypothesis

Published online by Cambridge University Press:  12 March 2014

David Lippel
David Lippel*
Affiliation:
Department of Mathematics., University of Notre Dame, Notre Dame. IN 46556-4618., USA, E-mail: dlippel@nd.edu
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Abstract

Let F be the class of complete, finitely axiomatizable ω-categorical theories. It is not known whether there are simple theories in F. We prove three results of the form: if T ∈ F has a sufficently well-behaved definable set J, then T is not simple. (In one case, we actually prove that T has the strict order property.) All of our arguments assume that the definable set J satisfies the Mazoyer hypothesis, which controls how an element of J can be algebraic over a subset of the model. For every known example in F, there is a definable set satisfying the Mazoyer hypothesis.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 2 , June 2005 , pp. 460 - 472
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1] Barwise, Jon, Admissible sets and structures, Springer-Verlag, Berlin, 1975.CrossRefGoogle Scholar
[2] Chatzidakis, ZoÉ and Hrushovski, Ehud, Model theory of difference fields, Transactions of the American Mathematical Society, vol. 351 (1999), no. 8, pp. 2997–3071.CrossRefGoogle Scholar
[3] Cherlin, Gregory and Hrushovski, Ehud, Finite structures with few types, Annals of Mathematics Studies, vol. 152, Princeton University Press, Princeton, NJ, 2003.Google Scholar
[4] Hodges, Wilfrid, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
[5] Ivanov, A. A., Countably categorical structures with a distributive lattice of algebraically closed subsets, Logic colloquium '92 (VeszprÉm, 1992), Studies in Logic, Language and Information, CSLI Publ., Stanford, CA. 1995, pp. 135–143.Google Scholar
[6] Ivanov, Alexandre A. and Macpherson, Dugald, Strongly determined types, Annals of Pure and Applied Logic, vol. 99 (1999), no. 1-3, pp. 197–230.CrossRefGoogle Scholar
[7] LindstrÖm, Per, On model-completeness, Theoriu (Lund), vol. 30 (1964), pp. 183–196.Google Scholar
[8] Lippel, David, Finitely axiomatizahle omega-categorical theories, Ph.D. thesis, University of California at Berkeley, 2001.Google Scholar
[9] Macpherson, Dugald, Finite axiomatizability and theories with trivial algebraic closure, Notre Dame Journal of Formal Logic, vol. 32 (1991), no. 2, pp. 188–192.CrossRefGoogle Scholar
[10] Mazoyer, Jacques, Sur les thÉories catÉgoriques finiment axiomatisables, Comptes Rendus Hebdomadaires des SÉances de l'AcadÉmie des Sciences, SÉries A et B, vol. 281 (1975), no. 12, pp. Ai, A403–A406.Google Scholar
[11] Onshuus, Alf, Thorn-forking in rosy theories, Ph.D. thesis, University of California at Berkeley, 2002.Google Scholar
[12] Onshuus, Alf, Properties and consequences of th-independence, (preprint, http://arxiv.org/abs/math.L0/0205004), 05 2002–5.Google Scholar
[13] Poizat, Bruno, Review of [16], Mathematical Reviews, MR 82m:03045.Google Scholar
[14] Saffe, JÜrgen, On categorical theories, Logic colloquium '84 (Manchester, 1984), Studies in Logic and the Foundations of Mathematics, vol. 120, North-Holland, Amsterdam, 1986, pp. 197–206.Google Scholar
[15] Shelah, Saharon, Classification theory and the number of nonisomorphie models, Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990.Google Scholar
[16] Zil'ber, B. I., Totally categorical theories: structural properties and the nonjinite axiomatizability, Model theory of algebra and arithmetic (Proc. conf., Karpacz, 1979), Lecture Notes in Mathematics, vol. 834, Springer, Berlin, 1980, pp. 381–410.CrossRefGoogle Scholar