Rich doctrines and Henkin's Theorem

Abstract: We find a possible interpretation of Henkin's Theorem in the language of existential implicational doctrines. Under some smallness assumption, starting from an implicational existential doctrine, with non-trivial fibers, we construct a new doctrine which is rich -- meaning that for every formula $\varphi(x)$ there is a constant $c$ such that $\exists x\varphi(x)$ has the same truth-value of $\varphi(c)$ -- and consistent. To obtain this result, we add a suitable amount of constants and axioms to the starting doctrine. We then show that a rich consistent doctrine admits an appropriate morphism towards the doctrine of subsets -- a model. Henkin's Theorem for doctrines follows from these two results, modeling our proof on the main lines of the original theorem.