## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 53, Issue 1 (1988), 231-242.

### A Downward Lowenheim-Skolem Theorem for Infinitary Theories which have the Unsuperstability Property

#### Abstract

We present a downward Lowenheim-Skolem theorem which transfers downward formulas from $L_{\infty,\omega}$ to $L_{\kappa^+,\omega}$. The simplest instance is: Theorem 1. Let $\lambda > \kappa$ be infinite cardinals, and let $L$ be a similarity type of cardinality $\kappa$ at most. For every $L$-structure $M$ of cardinality $\lambda$ and every $X \subseteq M$ there exists a model $N \prec M$ containing the set $X$ of power $|X| \cdot \kappa$ such that for every pair of finite sequences $\mathbf{a, b} \in N$ $\langle N, \mathbf{a}\rangle \equiv_{\| N \|^+,\omega} \langle N, \mathbf{b}\rangle \Leftrightarrow \langle M, \mathbf{a}\rangle \equiv_{\infty,\omega} \langle M, \mathbf{b}\rangle.$ The following theorem is an application: Theorem 2. Let $\lambda < \kappa, T \in L_{\kappa^+,\omega}$, and suppose $\chi$ is a Ramsey cardinal greater than $\lambda$. If $T$ has the $(\chi, L_{\kappa^+,\omega}$-unsuperstability property, then $T$ has the $(\chi, L_{\lambda^+,\omega})$-unsuperstability property.

#### Article information

**Source**

J. Symbolic Logic, Volume 53, Issue 1 (1988), 231-242.

**Dates**

First available in Project Euclid: 6 July 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1183742578

**Mathematical Reviews number (MathSciNet)**

MR929388

**Zentralblatt MATH identifier**

0646.03032

**JSTOR**

links.jstor.org

#### Citation

Grossberg, Rami. A Downward Lowenheim-Skolem Theorem for Infinitary Theories which have the Unsuperstability Property. J. Symbolic Logic 53 (1988), no. 1, 231--242. https://projecteuclid.org/euclid.jsl/1183742578