(en) Finite cardinals in topoi are introduced without assuming the existence of a natural number object. It is proved that this notion is equivalent to the conjunction of K-finiteness and simple finiteness. In this way, simple finiteness is viewed as a notion which corrects some defects of K-finiteness to obtain cardinal finiteness.
Mawanda, M.-M. (1988). Finite Cardinals in General Topoi. Journal of Pure and Applied Algebra, 50(2), 185-194. https://doi.org/10.1016/0022-4049(88)90114-4 (Original work published 1988)