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 $(N, 0, \pi)$, where the predecessor relation $\pi$ 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.