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Uncountable theories that are categorical in a higher power

Published online by Cambridge University Press:  12 March 2014

Michael Chris Laskowski
Michael Chris Laskowski*
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
*
Department of Mathematics, M.I.T., Cambridge, Massachusetts 02139
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Abstract

In this paper we prove three theorems about first-order theories that are categorical in a higher power. The first theorem asserts that such a theory either is totally categorical or there exist prime and minimal models over arbitrary base sets. The second theorem shows that such theories have a natural notion of dimension that determines the models of the theory up to isomorphism. From this we conclude that I(T,ℵα,) = ℵ0 + ∣α∣ where ℵα = the number of formulas modulo T-equivalence provided that T is not totally categorical. The third theorem gives a new characterization of these theories.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 2 , June 1988 , pp. 512 - 530
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[A1] Andler, D., Semi-minimal theories and categoricity, this Journal, vol. 40 (1975), pp. 419–438.Google Scholar
[A2] Andler, D., Finite-dimensional models of semi-minimal theories, Logique et Analyse, Nouvelle Serie, vol. 18 (1975), pp. 359–378.Google Scholar
[B-L] Baldwin, J. and Lachlan, A., On strongly minimal sets, this Journal, vol. 36 (1971), pp. 79–96.Google Scholar
[B1] Buechler, S., The geometry of weakly minimal types, this Journal, vol. 50 (1985), pp. 1044–1053.Google Scholar
[B2] Buechler, S., locally modular theories of finite rank, Annals of Pure and Applied Logic, vol. 30 (1986), pp. 83–94.CrossRefGoogle Scholar
[B3] Buechler, S. “Geometrical” stability theory, Logic Colloquium '85, North-Holland, Amsterdam, 1987, pp. 53–66.CrossRefGoogle Scholar
[E] Erimbetov, M. M., Complete theories with l-cardinal formulas, Algebra i Logika, vol. 14 (1975), pp. 245–257; English translation, Algebra and Logic , vol. 14 (1975), pp. 151–158.Google Scholar
[H1] Hrushovski, E., Locally modular regular types, Classification theory (Proceedings, Chicago, 1985), Springer-Verlag, Berlin (to appear).Google Scholar
[H2] Hrushovski, E., Contributions to stable model theory, Ph.D. thesis, University of California, Rerkeley, California, 1986.Google Scholar
[Mak] Makkai, M., A survey of basic stability theory with particular emphasis on orthogonality and regular types, Israel Journal of Mathematics, vol. 49 (1984), pp. 184–238.CrossRefGoogle Scholar
[Mar] Marsh, W., On ω1-categorical but not ω-categorical theories, Ph.D. thesis, Dartmouth College, Hanover, New Hampshire, 1966.Google Scholar
[Mor] Morley, M., Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514–538.CrossRefGoogle Scholar
[P] Pillay, A., An introduction to stability theory, Oxford University Press, Oxford, 1983.Google Scholar
[Sa] Saffe, J., One theorem on the number of uncountable models, Preprint No. 31, Institute of Mathematics, Siberian Branch Academy of Sciences of the USSR, Novosibirsk, 1983.Google Scholar
[Sh1] Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[Sh2] Shelah, S., Stability, the f. c. p., and superstability: model theoretic properties of formulas in first order theory, Annals of Mathematical Logic, vol. 3 (1971), pp. 271–362.CrossRefGoogle Scholar
[Z] Zil'ber, B. I., Strongly minimal countably categorical theories, II, III, Sibirskiĭ Matematicheskiĭ Zhurnal, vol. 25 (1984), no. 3, pp. 71–88; no. 4, pp. 63–77; English translation in Siberian Mathematical Journal , vol. 25 (1984), pp. 396–412, 559–571.Google Scholar