非良基集合的外延公理(英文)

The Axioms of Extensionality of Non-Well-Founded Sets

近年来,由于非良基集合在人工智能、认知科学及哲学等领域都有很重要的应用,它的研究越来越受到人们的关注。判断两个对象的同一性是集合论中最基本的问题,然而,与良基集合不同的是,非良基集合难以找到其最基本的组成成分,这样通常的外延公理就无法判断两个非良基集合(例如x={x}和y={y})相等。为了找到判断两个非良基集合相等的标准,我们必须强化通常的外延公理。利用Aczel四种非良基公理(AFA,SAFA,FAFA和BAFA),我们推出了四种判断两个非良基集合相等的标准,并且举例说明对于给定的两个非良基集合,如何判断它们相等,从而解决"循环集合"相等的问题。此外,笔者进一步论证判断这四种非良基集合相等的标准是通常外延公理的扩张,而不是替代。为此,本文首先给出了集合和图的一些基本定义和结果;其次讨论了由四种非良基公理AFA,SAFA,FAFA和BAFA分别确定的四种集合全域A,S,F和B;最后,讨论了外延公理的扩张。

In recent years, since non-well-founded sets have important applications in artificial intelligence, cognitive science and philosophy, there has been a furry of interest in them. How to judge the identity of two objects is the fundamental question in set theory. However, contrary to well founded sets, it is difficult to find the fundamental dements of non-well- founded sets. Thus the ordinary axiom of extensionality can not determine the equality of two non-well-founded sets, for example, the two sets x = {x} andy = {y}. In order to find a criterion to determine the equality of sets involving non-well-founded sets, we need something stronger than the ordinary axiom of extensionality. According to the four non-well-founded axioms （AFA, SAFA, FAFA, and BAFA） introduced by Aczel, we deduce four criteria for equality of non-well-founded sets. Furthermore, we illustrate how to use our criteria to determine the equality for two given non-well-founded sets by examples, so such problem of equality between ＂circular sets＂ has been solved. In addition, we also account for that the four criteria are the extensions of the ordinary axiom of extensionality, not replacement of it, To this end, in this paper we first present some basic definitions and results of sets and graph; secondly, we discuss the four set universes A, S, F, and B determined respectively by four non-well-founded axioms AFA, SAFA, FAFA, and BAFA; lastly, we study the extension of the ordinary axiom of extensionality.