Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 16F / i

This class is not axiomatisable. Suppose that a theory $T$ axiomatized it. Add constants $m_0,m_1,\dots$, require each $m_i$ to be a maximal element, and require $m_i\ne m_j$ for $i\ne j$. Every finite part of this enlarged theory has a model: choose a finite partially ordered set with enough maximal elements. By the compactness theorem the whole theory has a model, but its reduct is a model of $T$ with infinitely many maximal elements, a contradiction. This is the compactness obstruction to axiomatizing finitely many maximal elements.