## Journal of Symbolic Logic

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

Rami Grossberg

#### 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

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

Mathematical Reviews number (MathSciNet)
MR929388

Zentralblatt MATH identifier
0646.03032

JSTOR