Let $T$ be the theory of linear orders with no endpoints and let $\mathcal{L}=\{<,c_0,c_1,\dots\}$ be the language that consists of a binary relation symbol and a countable amount of constant symbols. I want to show that $T_\infty=T\cup\{c_n<c_{n+1}:n\in\mathbb{N}\}$ is complete. So far, most of the times I've showed that a theory is complete by proving that it's $\omega$-categorical. However, this approach won't work in this particular example. Apparently, this has been asked before in MSE, but the suggested arguments involve Ehrenfeucht-Fraïssé games, which I'm not familiar with.
Is there another way to show this? If not, can you explain how could I use EF games to solve this problem?
Any help will be highly appreciated.
To show $T$ is complete, we need to show that for every sentence $\varphi$, $T\models \varphi$ or $T\models\lnot\varphi$.
Observation: The sentence $\varphi$ only uses finitely many symbols, so it is is an $\mathcal{L}'$-sentence for some finite sublanguage $\mathcal{L'}\subseteq \mathcal{L}$.
Can you show that the restriction of $T$ to $\mathcal{L}'$-sentences is complete?