Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-30T22:52:07.664Z Has data issue: false hasContentIssue false
  • English
  • Français

On automorphism groups of countable structures

Published online by Cambridge University Press:  12 March 2014

Su Gao
Su Gao*
Affiliation:
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90024, USA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Strengthening a theorem of D. W. Kueker, this paper completely charaterizes which countable structures do not admit uncountable Lω1ω-elementarily equivalent models. In particular, it is shown that if the automorphism group of a countable structure M is abelian, or even just solvable, then there is no uncountable model of the Scott sentence of M. These results arise as part of a study of Polish groups with compatible left-invariant complete metrics.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 3 , September 1998 , pp. 891 - 896
Copyright
Copyright © Association for Symbolic Logic 1998

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.)

References

REFERENCES

[1]Barwise, J., Admissible sets and structures: an approach to definability theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1975.CrossRefGoogle Scholar
[2]Becker, H., Vaught's conjecture for complete left-invariant Polish groups, handwritten notes, 1996.Google Scholar
[3]Becker, H. and Kechris, A. S., The descriptive set theory of Polish group actions, London Mathematical Society Lecture Notes Series, no. 232, Cambridge University Press, Cambridge, 1996.CrossRefGoogle Scholar
[4]Hjorth, G. and Solecki, S., Vaught's conjecture and the Glimm-Effros property for Polish transformation groups, preprint, 1995.Google Scholar
[5]Hodges, W., Model theory, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
[6]Kechris, A. S., Classical descriptive set thoery, Graduate Texts in Mathematics, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
[7]Keisler, H. J., Model theory for infinitary logic, Studies in Logic and Foundations of Mathematics, vol. 62, North-Holland, 1971.Google Scholar
[8]Kueker, D. W., Definability, automorphisms and infinitary laguages, The syntax and semantics of infinitary languages (Barwise, J., editor), Lecture Notes in Mathematics, vol. 72, Springer-Verlag, 1968, pp. 152–165.CrossRefGoogle Scholar
[9]Makkai, M., An example concerning Scott heights, this Journal, vol. 46 (1981), pp. 301–318.Google Scholar
[10]Shelah, S., Tuuri, H., and Väänänen, J., On the number of automorphisms of uncountable models, this Journal, vol. 58 (1993), pp. 1402–1418.Google Scholar