Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 16F / v / 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 cardinalThe 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.