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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 ∈ N and a function s : N → 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 : O → 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.

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