= Solution
This class is not axiomatisable. It contains an infinite model, for example the <total order> $\mathbb Q$. If a theory $T$ in any fixed expanded language axiomatized it, choose a cardinal
$$
\kappa>\max\{|L|,|\mathbb R|\}.
$$
The <Upward Lowenheim-Skolem theorem> would give a model of $T$ of cardinality at least $\kappa$. Its underlying order cannot be isomorphic to a subset of $\mathbb R$, because every such subset has cardinality at most $|\mathbb R|$. This contradiction proves the claim.
Back to article page
