  We describe a theory of finite sets, and investigate the analogue of
Dedekind's theory of natural number systems (simply infinite systems) in this
theory. Unlike the infinitary case, in our theory, natural number systems come
in differing lengths and with different closure properties. We give examples of
natural number systems incomparable in length; we define hierarchies of natural
number systems closed under increasingly powerful functions; and we describe a
method by which to construct natural number systems with given closure
properties. These natural number systems form natural models for various
systems of weak arithmetic.
