Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-22T20:27:49.886Z Has data issue: false hasContentIssue false
  • English
  • Français

Indiscernible sequences in a model which fails to have the order property

Published online by Cambridge University Press:  12 March 2014

Rami Grossberg
Rami Grossberg*
Affiliation:
Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213
Get access Rights & Permissions [Opens in a new window]

Abstract

Basic results on the model theory of substructures of a fixed model are presented. The main point is to avoid the use of the compactness theorem, so this work can easily be applied to the model theory of Lω1,ω and its relatives. Among other things we prove the following theorem:

Let M be a model, and let λ be a cardinal satisfying λ∣L(M)∣ = λ. If M does not have the ω-order property, then for every A ⊆ M, ∣A∣ ≤ λ, and every I ⊆ M of cardinality λ+ there exists J ⊆ I cardinality λ+ which is an indiscernible set over A.

This is an improvement of a result of S. Shelah.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 1 , March 1991 , pp. 115 - 123
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.)

References

REFERENCES

[Gr]Grossberg, R., On chains of relatively saturated submodels of a model without the order property, this Journal, vol. 56(1991), pp. 124–128.Google Scholar
[GrSh]Grossbery, R. and Shelah, S., The number of nonisomorphic models of an infinitary theory which has the order property. Part A, this Journal, vol. 51 (1986), pp. 302–322.Google Scholar
[Ma]Marcus, L., A prime minimal model with an infinite set of indiscernibles, Israel Journal of Mathematics, vol. 11 (1972), pp. 180–183.CrossRefGoogle Scholar
[Sh1]Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[Sh2]Shelah, S., Universal classes. PartI, Classification theory (Baldwin, J. T., editor), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, Berlin, 1988, pp. 264–418.CrossRefGoogle Scholar
[Sh3]Shelah, S., A combinatorial program: stability and order for models and theories in infinitary languages, Pacific Journal of Mathematics, vol. 41 (1972), pp. 247–261.CrossRefGoogle Scholar
[Sh4]Shelah, S., Monadic logic: Hanf numbers, Around classification theory of models, Lecture Notes in Mathematics, vol. 1182, Springer-Verlag, Berlin, 1987, pp. 203–223.CrossRefGoogle Scholar