数理逻辑教材_悖论与数理逻辑的发展探析

发布时间：2016-10-26 11:02

  本文关键词：悖论与数理逻辑的发展探析，由笔耕文化传播整理发布。

悖论与数理逻辑的发展探析

     2009-7-31 11:09:13     来源：     

　　论文标题悖论与数理逻辑的发展探析
A Study on the Development of Paradox and Mathematical Logic
论文作者张莉敏
论文导师李娜,论文学位硕士,论文专业逻辑学
论文单位河南大学,点击次数1,论文页数39页File Size155k
2003-05-01悖论;数学基础;数理逻辑
paradox;mathematical foundation;mathematical logic
两千多年来，悖论一直是倍受逻辑学家关注的热点话题。在西方逻辑史上，曾有过三次悖论研究的高潮，尤其是罗素悖论所引发的第三次高潮，，直接促进了数理逻辑的形成和发展。这是因为罗素悖论的出现造成数学基础的危机，在循着如何排除悖论的思路进行数学基础研究所取得成果的基础上，数理逻辑中相继出现了三个划时代的成就，从而推动了数理逻辑的主要分支“四论”的产生和发展。本文分三部分对悖论与数理逻辑的关系进行了探讨，具体内容如下：第一部分主要论述了罗素悖论的出现及其影响，并对罗素悖论为何会造成数学基础的危机进行了具体分析。第二部分着重论述悖论是如何促进了数理逻辑的形成和发展。本文对这个问题从三个方面进行了分析：（一）悖论与数理逻辑三大学派的关系：其中对罗素的类型论进行了重点分析，并加入自己的思考。同时，对悖论如何促进直觉主义和形式主义学派的形成也进行了探讨。（二）悖论与数理逻辑三大成就的关系：其中以悖论与哥德尔不完全性定理的关系为重点，从悖论对哥德尔不完全性定理产生、构造及证明过程的影响进行了论证，并尝试做一些符号化等技术性的工作。此外，本文还对悖论与塔尔斯基的语义学和图灵机理论的关系进行了分析和研究。（三）论证了悖论在数理逻辑的主要分支“四论”的形成和发展中的作用。其中重点分析了公理化集合论，指出：它是为解决悖论问题而产生的，也是目前解决集合论悖论问题最好的方案。另外，数理逻辑的其它三个分支即证明论、递归论、和模型论也都是在研究悖论问题中逐渐形成和发展的。第三部分论述在探析悖论与数理逻辑的关系中所得到的意义和启示：只要我们把形式化的方法和哲学性的分析结合起来，用辩证的观点看问题，用系统的方法研究问题，悖论不但可以得到相对的解决，而且在解决悖论的过程中会引出一系列的重大发现。
Paradox has always been the central topic of the logicians for over 2000 years. There had been three climaxes in investigating paradox in the logical history of the west. Especially the appearance of Rusell"s paradox leads to the third climax of the research into the paradox, which has directly accelerated the formation and development of mathematical logic. The appearance of Russell"s paradox causes the crisis of mathematical foundation. Along the thinking way how we can dissolve paradox and on the basis of researches into the mathematical foundation, there are three epoch-making achievements of mathematical logic coming into view in succession, which accelerates the generation and development of Four Theories-main branches of mathematical logic. This paper probes into the relationship between paradox and mathematical logic. It consists of three parts, whose contents are as follows:The first part makes a main introduction to the appearance of Russell"s paradox, its influence, and analyzes how the Russell"s paradox causes the crisis of mathematical foundation.The second part makes the core of the essay, which is contributed to how the paradox accelerates the formation and development of mathematical logic. It falls into 3 sections: The first section is concerned with paradox and three schools of thought. It puts emphasis on the analysis of Russell"s theory of types, and follows with the author thought about it, at the same time, this section analyzes how the paradox promotes the formation of the school of intuitionism and of formalism. The second section is about paradox and three major achievements of mathematical logic. It centers on the relationship between Godel"s incompleteness theorem and the paradox. The author expounds the influences which paradox exerts on Godel"s Incompleteness Theorem"s generating, constructing, and the process of proof, and tries doing some technical work, such as symbolizing, etc. as well. In addition, the author analyzes and studies the<WP=5>relationship between paradox and Tagski"s Senantics and Theory of Turing Machine. The third section deals with the paradox as well as the formation and development of Four Theories ―main branch of mathematical logic. It concentrates on Theory of Axiomatic Sets, which comes into being with the resolution to the problem of paradox, and so do Theory of Recursive, Theory of Proof and Theory of Model. At present, the way of Theory of Axiomatic Sets is the best one to dissolve the paradox. The third part touches upon the enlightenments gained from the study on the relationship between paradox and mathematical logic on the relationship between paradox and mathematical logic, which are the following: so long as we combine the way of formalization with the philosophizing analysis, look at things dialectically, and deal with things systematically, not only can the problem of paradox be solved relatively, but also a series of important discovery can be found in the process of resolution to it.

  本文关键词：悖论与数理逻辑的发展探析，由笔耕文化传播整理发布。

本文编号：153922