数学逻辑：关于数字、集合、结构和对称性

English | PDF,EPUB | 2018 | 188 Pages | ISBN : 3319972979 | 7.11 MB

This book, presented in two parts, offers a slow introduction to mathematical logic, and several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions. Its first part, Logic Sets, and Numbers, shows how mathematical logic is used to develop the number structures of classical mathematics. The exposition does not assume any prerequisites; it is rigorous, but as informal as possible. All necessary concepts are introduced exactly as they would be in a course in mathematical logic; but are accompanied by more extensive introductory remarks and examples to motivate formal developments.

The second part, Relations, Structures, Geometry, introduces several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions, and shows how they are used to study and classify mathematical structures. Although more advanced, this second part is accessible to the reader who is either already familiar with basic mathematical logic, or has carefully read the first part of the book. Classical developments in model theory, including the Compactness Theorem and its uses, are discussed. Other topics include tameness, minimality, and order minimality of structures. The book can be used as an introduction to model theory, but unlike standard texts, it does not require familiarity with abstract algebra. This book will also be of interest to mathematicians who know the technical aspects of the subject, but are not familiar with its history and philosophical background.

这本书分为两部分，提供了一个慢速的数学逻辑入门，并介绍了一些模型论中的基本概念，如一阶可定义性、类型、对称性和初等扩展。 第一部分“逻辑、集合和数”展示了如何使用数学逻辑来发展经典数学的数结构。说明不依赖任何先修知识；它既严谨又尽可能地非正式。所有必要的概念都是按照在数学逻辑课程中引入的方式被精确介绍的，但都配以更详细的说明和示例，以促进形式发展的理解。 第二部分“关系、结构、几何”，介绍了模型论中的几个基本概念，如一阶可定义性、类型、对称性和初等扩展，并展示了这些是如何用于研究和分类数学结构。尽管更为深入，但这部分对于已经熟悉基本数学逻辑，或者已经仔细阅读了本书第一部分的读者来说是可理解的。包括核心定理在内的模型论中的经典发展也被讨论。其他话题还包括结构的坦率性、最小性和序量最小性。这本书可以作为模型论入门，但它不同于标准教科书，不需要熟悉抽象代数的知识。这本书也将对那些了解该领域的技术细节但不了解其历史和哲学背景的数学家感兴趣。

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