## Illinois Journal of Mathematics

### A non structure theorem for an infinitary theory which has the unsuperstability property

#### Abstract

Let $\kappa$, $\lambda$ be infinite cardinals, $\psi \in L_{\kappa^{+},\omega}$. We say that the sentence $\psi$ has the $\lambda$-unsuperstability property if there are $\{\varphi_{n}(\bar{\mathbf{x}},\bar{\mathbf{y}}): n < \omega\}$ quantifier free first order formulas in $L$, a model $M$ of $\psi$, and there exist $\{\bar{\mathbf{a}}_{\eta}: \eta \in^{\omega \geq} \lambda\} \subseteq |M|$ such that for all $\eta \in^{\omega}\lambda$, and for every $\nu \in^{\omega >}\lambda$, $$\nu < \eta \Leftrightarrow M \vDash \varphi_{l(\nu)}[\bar{\mathbf{a}}_{\nu},\bar{\mathbf{a}}_{\eta}].$$

#### Article information

Source
Illinois J. Math., Volume 30, Issue 2 (1986), 364-390.

Dates
First available in Project Euclid: 20 October 2009

https://projecteuclid.org/euclid.ijm/1256044645

Digital Object Identifier
doi:10.1215/ijm/1256044645

Mathematical Reviews number (MathSciNet)
MR840135

Zentralblatt MATH identifier
0578.03020

#### Citation

Grossberg, Rami; Shelah, Saharon. A non structure theorem for an infinitary theory which has the unsuperstability property. Illinois J. Math. 30 (1986), no. 2, 364--390. doi:10.1215/ijm/1256044645. https://projecteuclid.org/euclid.ijm/1256044645