Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 16J / c / Solution

If a theory axiomatized precisely the Archimedean fields, add a constant $c$ and sentences $t_n<c$ for every $n$. Every finite subset has an Archimedean model with $c$ sufficiently large, so compactness gives a model of the whole set. Its field reduct satisfies the proposed theory but is non-Archimedean, a contradiction.