Failed example of Downward Lowenheim Skolem Theorem in infinitary logic

Question

Find an example in a $\mathcal{L}_{\omega_1, \omega}$-theory $T$ ($\mathcal{L}$ is a countable language) such that every model of $T$ has cardinality at least $2^{\aleph_0}$. This can show that the Downward Lowenheim-Skolem theorem fails in infinitary logic.

I know that the Downward Lowenheim-Skolem theorem holds for first-order logic in which each sentence is of finite length. In this case, it means that every first-order theory with an uncountable model also has a countable model.

Answer 1

Let $\mathcal L$ consist of a constant symbol $q$ for each rational number and a relation symbol $<.$ The axioms are

1. The first-order axioms saying $<$ is a strict linear order
2. For each real number $r,$ the axiom $\exists! x\bigwedge_q \phi_q$ where $\phi_q$ says either $q<x$ or $x<q$ or $q = x$, whichever of the three holds for $x=r$.