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In this paper, we give an axiomatization of the ordinal number system, in thestyle of Dedekind's axiomatization of the natural number system. The latter isbased 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 axiomatizationis 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 $s\colon O\to O$. Infact, we develop the theory relative to a Grothendieck-style universe (minusthe power-set axiom), as a way of bringing the natural and the ordinal casesunder one framework. We also establish a universal property for the ordinalnumber system, analogous to the well-known universal property for the naturalnumber system.

A Dedekind-style axiomatization and the corresponding universal property of an ordinal number system