Classifying topoi and the axiom of infinity

Let 6 e be an elementary topos. The axiom of infinity, asserting that 5 e has a natural numbers object, is shown to be necessary-sufficiency has long been k n o w n f o r the existence of an object-classifying topos over 5 e. In the known constructions [1, 4, 6] of classifying topoi for geometric theories, it is always assumed that the base topos fie satisfies the axiom of infinity. The purpose of this note is to show that this assumption is necessary. We show that, if a certain very simple geometric theory T has a classifying topos over fie, then fie has a natural numbers object. We also show, using well-known methods [1, 4, 6], that the existence o f a classifying topos for T follows from the existence of an object classifier, i.e., a classifying topos for the one-sorted geometric theory having no non-logical symbols and no axioms, the theory whose models in any category are just the objects of that category. Thus, our main result, which answers a question posed in [5], can be stated as follows. T H E O R E M . Suppose that there exists an object classifier over the elementary topos fie. Then fie has a natural numbers object. Our notation and terminology will be standard [4]. Let T be the one-sorted geometric theory having one constant symbol o, one unary function symbol ~, no other non-logical symbols, and no axioms. Thus, a T-model in any topos (or in any category with a terminal object 1) is simply a structure ~4 = (A, oa, ~a) where A is an object and oa : 1--->A and~a :A--->A. (We shall omit the subscripts on o and ~ whenever no confusion can arise). A homomorphism of T-models is a morphism between the underlying objects that commutes with the o and ~ morphisms. A natural numbers object is simply an initial T-model. A weakly initial T-model (in any topos), i.e., a T-model having Presented by F. E. J. Linton. Received October 30, 1987 and in final form February 24, 1988.