Axiomatizations of natural numbers

We introduce two axiomatizations of natural numbers and place them in the context of the well-known formalizations of natural numbers by Frege, Dedekind, Peano, Russell, and Devide. To this end, we are developing a methodology and notations that allow a uniform presentation of these different formalizations. We prove that our axiomatizations categorically axiomatize the structure (N, 0, ?), where the predecessor relation ? can be the immediate predecessor p or the general predecessor <. The first three axioms for the immediate and general predecessor are exactly the same, but the fourth axioms are specific for p and <. One postulates that the inverse of the immediate predecessor is a function, the other that the general predecessor is a total relation. We do not postulate that the inverse is an injection or that < is an order. Finally, we discuss Henkin?s analysis of Peano?s axiomatization in the same context.