Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Uncountable categoricity for gross models


Authors: Michael C. Laskowski and Anand Pillay
Journal: Proc. Amer. Math. Soc. 132 (2004), 2733-2742
MSC (2000): Primary 03C45; Secondary 03C50, 03C75
DOI: https://doi.org/10.1090/S0002-9939-04-07451-9
Published electronically: March 25, 2004
MathSciNet review: 2054800
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A model $M$is said to be gross if all infinite definable sets in $M$ have the same cardinality (as $M$). We prove that if for some uncountable $\kappa$, $T$ has a unique gross model of cardinality $\kappa$, then for any uncountable $\kappa$, $T$ has a unique gross model of cardinality $\kappa$.


References [Enhancements On Off] (What's this?)

  • 1. J. T. Baldwin, Fundamentals of Stability Theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1988. MR 89k:03002
  • 2. J. T. Baldwin and A. H. Lachlan, On strongly minimal sets, Journal of Symbolic Logic 36 (1971), 79-96. MR 44:3851
  • 3. G. Cherlin, L. A. Harrington, and A. H. Lachlan, $\aleph_0$-categorical, $\aleph_0$-stable structures, Annals of Pure and Applied Logic 28 (1985), 103-135. MR 86g:03054
  • 4. H. J. Keisler, Model theory for infinitary logic. Logic with countable conjunctions and finite quantifiers, North-Holland, Amsterdan, 1971. MR 49:8855
  • 5. B. Kim, Simple first order theories, Ph.D. thesis, University of Notre Dame, 1996.
  • 6. M. Morley, Categoricity in power, Trans. Amer. Math. Soc. 114 (1965), 514-538. MR 31:58
  • 7. R. Moosa, Contributions to the model theory of fields and compact complex spaces, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2001.
  • 8. A. Pillay, Geometric Stability Theory, The Clarendon Press, Oxford University Press, New York, 1996. MR 98a:03049
  • 9. S. Shelah, The lazy model-theoretician's guide to stability, Logique et Analyse (N.S.) 18 (1975). MR 58:27447
  • 10. S. Shelah, Simple unstable theories, Annals of Mathematical Logic 19 (1980), 177-203. MR 82g:03055
  • 11. S. Shelah, Classification Theory and the Number of Nonisomorphic Models, 2nd ed., North-Holland, Amsterdam, 1990. MR 91k:03085

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03C45, 03C50, 03C75

Retrieve articles in all journals with MSC (2000): 03C45, 03C50, 03C75


Additional Information

Michael C. Laskowski
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: mcl@math.umd.edu

Anand Pillay
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: pillay@math.uiuc.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07451-9
Received by editor(s): June 9, 2003
Published electronically: March 25, 2004
Additional Notes: The first author was partially supported by NSF grant DMS-0071746
The second author was partially supported by NSF grants DMS-0070179 and DMS 01-00979 and a Humboldt Foundation Research Award
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society