论文标题
Dedekind风格的公理化和序数系统的相应通用属性
A Dedekind-style axiomatization and the corresponding universal property of an ordinal number system
论文作者
论文摘要
在本文中，我们以Dedekind的自然数量系统的公理化方式对序数系统进行公理化。后者基于结构$（n，0，s）$，该结构由套装$ n $，n $中的杰出元素$ 0 \和函数$ s \ colon n \ to n $组成。我们的Axiomatization中的结构是三重$（o，l，s）$，其中$ o $是一类，$ l $是$ o $的所有$ s $ clucted的“ subsets'的函数，而$ s $是类函数$ s \ s \ colon o \ colon o \ o $。实际上，我们发展了与Grothendieck风格的宇宙（减去电源公理）相关的理论，作为一种在一个框架下将自然案例和顺序案例带入的一种方式。我们还为序数系统建立了通用属性，类似于自然数系统的众所周知的通用属性。
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.
