## Journal of Symbolic Logic

### On Chains of Relatively Saturated Submodels of a Model Without the Order Property

Rami Grossberg

#### Abstract

Let $M$ be a given model with similarity type $L = L(M)$, and let $L'$ be any fragment of $L_{|L(M)|}^+, \omega$ of cardinality $|L(M)|$. We call $N \prec M L'$-relatively saturated $\operatorname{iff}$ for every $B \subseteq N$ of cardinality less than $\| N \|$ every $L'$-type over $B$ which is realized in $M$ is realized in $M$ is realized in $N$. We discuss the existence of such submodels. The following are corollaries of the existence theorems. (1) If $M$ is of cardinality at least $\beth_{\omega_1}$, and fails to have the $\omega$ order property, then there exists $N \prec M$ which is relatively saturated in $M$ of cardinality $\beth_{\omega_1}$. (2) Assume GCH. Let $\psi \in L_{\omega_1, \omega$, and let $L' \subseteq L_{\omega 1, \omega$ be a countable fragment containing $\psi$. If $\exists \chi > \aleph_0$ such that $I(\chi, \psi) < 2^\chi$, then for every $M \models \psi$ and every cardinal $\lambda < \|M\|$ of uncountable cofinality, $M$ has an $L'$-relatively saturated submodel of cardinality $\lambda$.

#### Article information

Source
J. Symbolic Logic, Volume 56, Issue 1 (1991), 124-128.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743556

Mathematical Reviews number (MathSciNet)
MR1131735

Zentralblatt MATH identifier
0733.03023

JSTOR