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

TL;DR

This paper develops a Dedekind-style axiomatization of the ordinal number system, unifying natural and ordinal numbers under a common framework and establishing a universal property similar to that of natural numbers.

Contribution

It introduces a novel axiomatization of ordinal numbers using a triple structure and proves a universal property for this system, extending Dedekind's approach.

Findings

A new Dedekind-style axiomatization of ordinal numbers.

Unified framework for natural and ordinal numbers.

Universal property of the ordinal number system established.

Abstract

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.