A Study On The Development Of Paradox And Mathematical Logic

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 therelationship 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.