Hostname: page-component-cb9f654ff-kl2l2 Total loading time: 0 Render date: 2025-08-21T19:33:32.099Z Has data issue: false hasContentIssue false
  • English
  • Français

On the strong Martin conjecture

Published online by Cambridge University Press:  12 March 2014

Masanori Itai
Masanori Itai*
Affiliation:
Department of Mathematics, St. Lawrence University, Canton, New York 13617
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the following conjecture.

Conjecture.Let T be an ω-stable theory with continuum many countable models. Then either i) T has continuum many complete extensions in L 1(T), or ii) some complete extension of T in L 1has continuum many L 1-types without parameters.

By Shelah's proof of Vaught's conjecture for ω-stable theories, we know that there are seven types of ω-stable theory with continuum many countable models. We show that the conjecture is true for all but one of these seven cases. In the last case we show the existence of continuum many L 2-types.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 3 , September 1991 , pp. 862 - 875
Copyright
Copyright © Association for Symbolic Logic 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[Bal] Baldwin, J. T., Fundamentals of stability theory, Springer-Verlag, New York, 1988.CrossRefGoogle Scholar
[Bar] Barwise, K. J., Admissible sets and structures, Springer-Verlag, New York, 1975.CrossRefGoogle Scholar
[Bous1] Bouscaren, E., Countable models of multidimensional ℵ0-stable theories, Israel Journal of Mathematics, vol. 48 (1983), pp. 377–383.Google Scholar
[Bous2] Bouscaren, E., Martin's conjecture for ω-stable theories, Israel Journal of Mathematics, vol. 49 (1984), pp. 15–26.CrossRefGoogle Scholar
[Buec] Buechler, S., Newelski's proof of Saffe's conjecture, Handwritten note, 1988.Google Scholar
[CK] Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[EFT] Ebbinghaus, H. D., Flum, J., and Thomas, W., Mathematical logic, Springer-Verlag, New York, 1984.Google Scholar
[G] Garavaglia, S., Decomposition of totally transcendental modules, this Journal, vol. 45 (1980), pp. 155–164.Google Scholar
[HM] Harrington, L. and Makkai, M., An exposition of Shelah's “Main Gap”: counting uncountable models of ω-stable and superstable theories, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 139–177.CrossRefGoogle Scholar
[I] Itai, M., On the strong Martin conjecture, Ph.D. Thesis, University of Illinois at Chicago, Chicago, Illinois, 1989.Google Scholar
[K] Keisler, H. J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.Google Scholar
[L] Lascar, D., Why some people are excited by Vaught's conjecture, this Journal, vol. 50 (1985), pp. 973–982.Google Scholar
[Ma] Makkai, M., A survey of basic stability theory, with particular emphasis on orthogonality and regular types, Israel Journal of Mathematics, vol. 49 (1984), pp. 181–238.CrossRefGoogle Scholar
[Mo] Morley, M., The number of countable models, this Journal, vol. 35 (1970), pp. 14–18.Google Scholar
[Ne] Newelski, L., A proof of Saffe's conjecture, Preprint, 1988.Google Scholar
[P] Pillay, A., An introduction to stability theory, Clarendon Press, Oxford, 1983.Google Scholar
[R] Rubin, M., Theories of linear order, Israel Journal of Mathematics, vol. 17 (1974), pp. 392–443.CrossRefGoogle Scholar
[S] Shelah, S., Classification theory and the number of nonisomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[SHM] Shelah, S., Harrington, L., and Makkai, M., A proof of Vaught's conjecture for ω-stable theories, Isreal Journal of Mathematics, vol. 49 (1984), pp. 259–278.CrossRefGoogle Scholar
[W] Wagner, C. W., On Martin's conjecture, Annals of Mathematical Logic, vol. 22 (1982), pp. 47–68.CrossRefGoogle Scholar