Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 120 / 3 / c / Solution

No such first-order theory exists. Suppose $T$ axiomatized the Heyting algebras having only finitely many regular elements. Expand the language by constants $c_n$ and addEvery finite subset has a model: take a sufficiently large finite Boolean algebra, in which every element is regular. By the compactness theorem, the entire expanded theory has a model. Its reduct is a model of $T$ with infinitely many distinct regular elements, contradicting the proposed axiomatization.