Definability in Infinitary Languages and Invariance by Automorphisms

Abstract

Given a $\mathcal L_{\alpha\beta} ^E$-structure $E$, where $\mathcal L_{\alpha\beta} ^E$ is an infinitary language, we show that $\alpha$ and $\beta$ can be chosen in such way that every orbit of the group $G$ of automorphisms of $E$ is $\mathcal L_{\alpha\beta} ^E$-definable. It follows that two sequences of elements of the domain $D$ of $E$ satisfy the same set of $\mathcal L_{\alpha\beta}$-formulas if and only if they are in the same orbit of $G$