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首页> 外文期刊>The bulletin of symbolic logic >The empty set, the singleton, and the ordered pair
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The empty set, the singleton, and the ordered pair

机译：空集，单例和有序对

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For the modern set theorist the empty set φ, the singleton {a}, and the ordered pair are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quiekcly assimilated long before the complexities of Power Set. Replacement, and Choice are broached in the formal elaboration of the 'set of' {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts severs as a motif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the literative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all.
机译：对于现代集合理论家来说，空集合φ，单例{a}和有序对处于集合论的系统化，公理化发展的开始，无论是作为数学领域还是作为统一框架对于正在进行的数学。这些概念是由Ernst Zermelo [1908a]的最初公理化发展而来的抽象抽象集概念中最简单的构成部分，并且早在Power Set的复杂性之前就已被同化。替换和选择在“ set of” {}操作的正式阐述中涉及。因此，令人惊讶的是，尽管这些概念在今天已不成问题，但它们曾一度引起数学逻辑领域的领先先驱者如Frege，Russell，Dedekind和Peano的关注和困惑。在摆脱19世纪逻辑动荡的现代数学逻辑的发展过程中，空集，单例和有序对的出现作为清晰而基本的集理论的概念不断演变，成为反映和启发更大，更重要的主题。数学逻辑的发展：从内涵观点到扩展观点的转变，类型区别的发展，集合的逻辑与文学的概念的融合以及各种概念和原理的出现，它们是集合论而非纯粹逻辑的。这里与塔斯基（Tarski）对形式语言的真理的递归定义有一个宽松的类比：数学兴趣主要在于递归的过程和模型理论中附带的形式语义，而哲学利益则主要在于递归，真理和行为的基础。在基本谓词一级的含义。绕回开始，我们将看看空集，单例和有序对到底有多重要。

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