I am asked to verify whether or not you can axiomatise the theory of countable dense total orderings in the language of partially ordered sets.
We have the Upward Lowenheim-Skolem Theorem, telling us that for a language $L$ and theory $S\subset L$ then if $S$ has an infinite model, it has an uncountable model.
My first thought was that if the theory of countable dense total orderings was axiomatisable, then $\mathbb Q$ with the usual ordering would an infinite model, hence by Upward Lowenheim-Skolem, there exists an uncountable model, and so we have a contradiction.
However, after more thought is this really a contradiction? Can't it be the case that "inside" of this uncountable model, it looks like its countable?
Is it perhaps the case then that the theory is indeed axiomatisable?