Classifying topoi and the axiom of infinity (Q1263664)

In the known constructions of a classifying topos for geometric theories, it is always assumed that the base topos \({\mathcal S}\) satisfies the axiom of infinity, asserting that \({\mathcal S}\) has a natural number object. The purpose of this note is to show that this assumption is necessary. It is indeed proved that the existence of an object classifying topos over \({\mathcal S}\) implies that a very simple geometric theory \({\mathbb{T}}\) has a classifying topos, and that the existence of this classifying topos implies the axiom of infinity.