Finite cardinals in general topoi (Q1103042)

In a topos with a natural number object the full subcategory of finite cardinals (as defined by J. Bénabou) is a subtopos. With a natural number object, K-finite objects (in the name of Kuratowski) have been investigated and it has been found out that the full subcategory of K- finite objects is not in general a topos. The aim of the paper is to define a notion of finite cardinal which coincides with the former notion if there is a natural number object in the topos and which in a general topos gives a subtopos satisfying the axiom of choice. These finite cardinals are K-finite objects which satisfy moreover the following condition: there exists a total order on the object which is a simple well ordering as well as its reverse order.