Mathematical logic - foundations for information science
Mathematical logic - foundations for information science
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DOI：
10.1007/978-3-7643-9977-1
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发表时间：
2010-01
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影响因子：
--
通讯作者：
Wei Li
中科院分区：
文献类型：
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作者：
Wei Li
经典数理逻辑被认为是数学基础的重要组成部分。它是对数学方法的研究，特别是公理系统的性质和证明的结构。数理逻辑的核心包括定义一阶语言的语法，研究它们的模型，形式化逻辑推理并证明其可靠性和完备性。它还涵盖了可计算性理论和哥德尔不完备性定理。这一抽象过程始于世纪末，到1950年基本完成。1990年，我开始讲授数理逻辑课程。这次教学经历使我认识到，虽然演绎逻辑被很好地分析了，但公理化的过程却没有得到深入的研究。几年后，我组织了一系列研讨会，作为随后的努力。前五次研讨会涵盖了经典的数理逻辑和其余的是一个初步纲要的正式理论的公理化。随着我对数理逻辑的理解越来越深入，我对公理化过程的分析和形式化的渴望也越来越强烈。我也看到了数理逻辑在信息技术和科学研究中的影响。这激发了我为生活在信息社会的学生写一本书的灵感。计算机发明于20世纪40年代，高级编程语言随后被定义和实现。从那时起，计算机科学迅速发展。这对数理逻辑产生了深远的影响，因为它的概念和理论得到了广泛的应用。然而，计算机科学的发展反过来又对数理逻辑提出了新的要求，这是我研究的重点，也是本书的动机。这一动机是由两个考虑因素指导的。
Classical mathematical logic is considered to be an important component of the foundation of mathematics. It is the study of mathematical methods, especially the properties of axiom systems and the structure of proofs. The core of mathematical logic consists of defining the syntax of first-order languages, studying their models, formalizing logical inference and proving its soundness and completeness. It also covers the theory of computability and Gödel’s incompleteness theorems. This process of abstraction started in the late 19th Century and was essentially completed by 1950. In 1990, I began to give courses on mathematical logic. This teaching experience made me realize that, although deductive logic was well analyzed, the process of axiomatization had not been studied in depth. Several years later, I organized a series of seminars as an ensuing effort. The first five seminars covered classical mathematical logic and the rest were a preliminary outline of the formal theory of axiomatization. As my understanding of mathematical logic became deeper, my desire to analyze and formalize the process of axiomatization became more intense. I also saw the influence of mathematical logic in information technology and scientific research. This inspired me to write a book for students living in the information society. The computer was invented in the 1940’s and high-level programming languages were defined and implemented soon afterwards. Computer science has developed rapidly since then. This exerted a profound influence on mathematical logic, because its concepts and theories were extensively applied. However, the development of computer science has, in turn, made new demands on mathematical logic, which have been the focus of my research and the motivation for this book. This motivation is guided by two considerations.