In this paper, we give an axiomatization of the ordinal number system, in the style of Dedekind's axiomatization of the natural number system. The latter is based on a structure \((N,0,s)\) consisting of a set \(N\), a distinguished element \(0\in N\) and a function \(s\colon N\to N\). The structure in our axiomatization is a triple \((O,L,s)\), where \(O\) is a class, \(L\) is a function defined on all \(s\)-closed `subsets' of \(O\), and \(s\) is a class function \(s\colon O\to O\). In fact, we develop the theory relative to a Grothendieck-style universe (minus the power-set axiom), as a way of bringing the natural and the ordinal cases under one framework. We also establish a universal property for the ordinal number system, analogous to the well-known universal property for the natural number system.