本文關(guān)鍵詞：悖論與數(shù)理邏輯的發(fā)展探析，由筆耕文化傳播整理發(fā)布。

悖論與數(shù)理邏輯的發(fā)展探析

     2009-7-31 11:09:13     來源：     

　　論文標(biāo)題悖論與數(shù)理邏輯的發(fā)展探析
A Study on the Development of Paradox and Mathematical Logic
論文作者張莉敏
論文導(dǎo)師李娜,論文學(xué)位碩士,論文專業(yè)邏輯學(xué)
論文單位河南大學(xué),點(diǎn)擊次數(shù)1,論文頁數(shù)39頁File Size155k
2003-05-01悖論;數(shù)學(xué)基礎(chǔ);數(shù)理邏輯
paradox;mathematical foundation;mathematical logic
兩千多年來，悖論一直是倍受邏輯學(xué)家關(guān)注的熱點(diǎn)話題。在西方邏輯史上，曾有過三次悖論研究的高潮，尤其是羅素悖論所引發(fā)的第三次高潮，，直接促進(jìn)了數(shù)理邏輯的形成和發(fā)展。這是因?yàn)榱_素悖論的出現(xiàn)造成數(shù)學(xué)基礎(chǔ)的危機(jī)，在循著如何排除悖論的思路進(jìn)行數(shù)學(xué)基礎(chǔ)研究所取得成果的基礎(chǔ)上，數(shù)理邏輯中相繼出現(xiàn)了三個劃時(shí)代的成就，從而推動了數(shù)理邏輯的主要分支“四論”的產(chǎn)生和發(fā)展。本文分三部分對悖論與數(shù)理邏輯的關(guān)系進(jìn)行了探討，具體內(nèi)容如下：第一部分主要論述了羅素悖論的出現(xiàn)及其影響，并對羅素悖論為何會造成數(shù)學(xué)基礎(chǔ)的危機(jī)進(jìn)行了具體分析。第二部分著重論述悖論是如何促進(jìn)了數(shù)理邏輯的形成和發(fā)展。本文對這個問題從三個方面進(jìn)行了分析：（一）悖論與數(shù)理邏輯三大學(xué)派的關(guān)系：其中對羅素的類型論進(jìn)行了重點(diǎn)分析，并加入自己的思考。同時(shí)，對悖論如何促進(jìn)直覺主義和形式主義學(xué)派的形成也進(jìn)行了探討。（二）悖論與數(shù)理邏輯三大成就的關(guān)系：其中以悖論與哥德爾不完全性定理的關(guān)系為重點(diǎn)，從悖論對哥德爾不完全性定理產(chǎn)生、構(gòu)造及證明過程的影響進(jìn)行了論證，并嘗試做一些符號化等技術(shù)性的工作。此外，本文還對悖論與塔爾斯基的語義學(xué)和圖靈機(jī)理論的關(guān)系進(jìn)行了分析和研究。（三）論證了悖論在數(shù)理邏輯的主要分支“四論”的形成和發(fā)展中的作用。其中重點(diǎn)分析了公理化集合論，指出：它是為解決悖論問題而產(chǎn)生的，也是目前解決集合論悖論問題最好的方案。另外，數(shù)理邏輯的其它三個分支即證明論、遞歸論、和模型論也都是在研究悖論問題中逐漸形成和發(fā)展的。第三部分論述在探析悖論與數(shù)理邏輯的關(guān)系中所得到的意義和啟示：只要我們把形式化的方法和哲學(xué)性的分析結(jié)合起來，用辯證的觀點(diǎn)看問題，用系統(tǒng)的方法研究問題，悖論不但可以得到相對的解決，而且在解決悖論的過程中會引出一系列的重大發(fā)現(xiàn)。
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.

  本文關(guān)鍵詞：悖論與數(shù)理邏輯的發(fā)展探析，由筆耕文化傳播整理發(fā)布。

本文編號：153922