= Solution
This class is also not first-order <axiomatisable>. Suppose such a theory had an uncountable dense-order model $M$. Its language is finite. Starting with a countably infinite set of elements and repeatedly adjoining witnesses for all existential formulas with parameters already chosen yields a countable elementary substructure: there are only countably many formulas and parameter tuples at every stage. The induction-on-formulas argument from part (i) proves elementarity. The substructure satisfies every sentence of the theory, yet is countable, a contradiction. This is the witness construction underlying the <Downward Lowenheim-Skolem theorem>.
$$
\boxed{\text{Uncountable dense total orders are not first-order axiomatizable.}}
$$
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