Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-28T19:16:37.471Z Has data issue: false hasContentIssue false
  • English
  • Français

Rich models

Published online by Cambridge University Press:  12 March 2014

Michael H. Albert  and
Rami P. Grossberg
Michael H. Albert
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Rami P. Grossberg
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Get access Rights & Permissions [Opens in a new window]

Abstract

We define a rich model to be one which contains a proper elementary substructure isomorphic to itself. Existence, nonstructure, and categoricity theorems for rich models are proved. A theory T which has fewer than min(2λ, ℶ2) rich models of cardinality λ (λ > ∣T∣) is totally transcendental. We show that a countable theory with a unique rich model in some uncountable cardinal is categorical in ℵ1 and also has a unique countable rich model. We also consider a stronger notion of richness, and use it to characterize superstable theories.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 3 , September 1990 , pp. 1292 - 1298
Copyright
Copyright © Association for Symbolic Logic 1990

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]Baldwin, J. T., The spectrum of resplendency, this Journal, vol. 55 (1990), pp. 626–636.Google Scholar
[2]Buechler, Steven, Resplendency and recursive definability in ω-stable theories, Israel Journal of Mathematics, vol. 49 (1984), pp. 26–33.CrossRefGoogle Scholar
[3]Shelah, Saharon, The lazy model-theoretician's guide to stability theory, Logique et Analyse (Nouvelle Série), vol. 18 (1975), pp. 241–308.Google Scholar
[4]Shelah, Saharon, Classification of non-elementary classes. II, Classification theory (proceedings, Chicago, 1985, Baldwin, J. T., editor), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, Berlin, 1987, pp. 419–497.Google Scholar
[5]Shelah, Saharon, Thenumber of pairwise non-elementarily-embeddable models, this Journal, vol. 54 (1989), pp. 1431–1456.Google Scholar
[6]Shelah, Saharon, Classification theory and the number of non-isomorphic models, 2nd ed., North-Holland, Amsterdam (to appear).Google Scholar