# All classes of finite structures are axiomatizable in $L_{\infty\omega}$

We want to proof that every class of finite structures is axiomatizable in the infinitary logic $L_{\infty\omega}$.

We fix the signature $\tau$ (is okay to do so?). Thus, we can assume that for every finite $\tau$-structure $c$ there is a finite formula $\varphi_c \in L_{\infty\omega}$ describing it. Now for every class $C$ of finite structures we could define

$\Phi := \{ \varphi_c \mid c \in C\}$

The problem is if we want to combine this class to a formula such as this

$\psi := \lor \Phi$.

We have to assume that $C$ is a set not just a class. At least that $\Phi$ is a set. Any hints?

Here is a sketch to show that $\Phi$ is a set. $\Phi \subset L_{\infty\omega}$ and $\psi(\varrho) := \exists c \in C (\varrho = \varphi_c)$.

$\Rightarrow \Phi = \{\varrho \in L_{\infty\omega} \mid \psi(\varrho)\}$. Thus by the Axiom of Seperation $\Phi$ is a set. Is that correct?

-
I guess you should reduce the problem to showing that there is at most a set of isomorphism classes... – Zhen Lin Jan 13 '13 at 15:51

You have the right idea, except that $L_{\infty \omega}$ is a proper class itself. However each finite structure is described upto isomorphism by an $L_{\omega \omega}$ formula. That is, in fact $\varphi_c \in L_{\omega \omega}$ and $\Phi \subseteq L_{\omega \omega}$.
Thanks a lot. Out of curiosity: Why is $L_{\infty\omega}$ a proper class? – joachim Jan 14 '13 at 20:50