重新考虑有序对

IF 0.7 3区数学 Q1 LOGIC

Bulletin of Symbolic Logic Pub Date : 2008-09-01 DOI:10.2178/bsl/1231081372

D. Scott, D. McCarty

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引用次数: 13

摘要

众所周知的Wiener-Kuratowski关于有序对的显式定义，即集合(x,y) = {{x}， {x,y}}，在许多集合理论中都是有效的，但对于那些类不能是单例元素的集合理论就不适用了。在基础公理的帮助下，我们提出了有序对的递归定义，它解决了这一缺点，并自然地推广到更长的有序元组。新定义有许多优点，因为它允许统一定义在集合理论的广泛模型中同样有效。在ZFC及其密切相关的理论中，在新的定义下，两个无限集合的有序对的秩等于这些集合的秩的最大值。

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Reconsidering Ordered Pairs

Abstract The well known Wiener-Kuratowski explicit definition of the ordered pair, which sets (x,y) = {{x}, {x,y}}, works well in many set theories but fails for those with classes which cannot be members of singletons. With the aid of the Axiom of Foundation, we propose a recursive definition of ordered pair which addresses this shortcoming and also naturally generalizes to ordered tuples of greater length. There are many advantages to the new definition, for it allows for uniform definitions working equally well in a wide range of models for set theories. In ZFC and closely related theories, the rank of an ordered pair of two infinite sets under the new definition turns out to be equal to the maximum of the ranks of the sets.

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来源期刊

Bulletin of Symbolic Logic 数学-数学

CiteScore

0.60

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The Bulletin of Symbolic Logic was established in 1995 by the Association for Symbolic Logic to provide a journal of high standards that would be both accessible and of interest to as wide an audience as possible. It is designed to cover all areas within the purview of the ASL: mathematical logic and its applications, philosophical and non-classical logic and its applications, history and philosophy of logic, and philosophy and methodology of mathematics.