Expressing “uncountable” in $L_{\omega_1\omega}$

Question

Given a countable signature $\tau$ I'm trying to find a uncountable $\tau$-Structure $\mathfrak{A}$ which does not satisfy the same infinitary logic $L_{\omega_1\omega}$-sentences as a countable $\tau$-structure $\mathfrak{B}$. This is basically a weaker version of the Exercise 1.7. My guess is to try to construct a power set construction with the help of a countable signature. Is this a good approach? Any other ideas?

Answer 1

HINT: Try giving $\mathscr{L}$ constant symbols $a_n$ for $n\in\omega$ and a binary relation symbol $E$, and for each $A\subseteq\omega$ letting $\varphi_A$ be

$$\exists x\left(\bigwedge_{n\in A}E(a_n,x)\land\bigwedge_{n\in\omega\setminus A}\neg E(a_n,x)\right)\;.$$

You’ll also want $a_m\ne a_n$ for $m,n\in\omega$ with $m\ne n$.

(And yes, your approach is entirely reasonable.)