## Journal of Symbolic Logic

### Indiscernible Sequences in a Model which Fails to have the Order Property

Rami Grossberg

#### 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 $L_{\omega_1,\omega}$ and its relatives. Among other things we prove the following theorem: Let $M$ be a model, and let $\lambda$ be a cardinal satisfying $\lambda^{|L(M)|} = \lambda$. If $M$ does not have the $\omega$-order property, then for every $A \subseteq M, |A| \leq \lambda$, and every $\mathbf{I} \subseteq M$ of cardinality $\lambda^+$ there exists $\mathbf{J} \subseteq \mathbf{I}$ of cardinality $\lambda^+$ which is an indiscernible set over $A$. This is an improvement of a result of S. Shelah.

#### Article information

Source
J. Symbolic Logic, Volume 56, Issue 1 (1991), 115-123.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743555

Mathematical Reviews number (MathSciNet)
MR1131734

Zentralblatt MATH identifier
0733.03022

JSTOR