摘要
从新构建的基础理论学的角度,讨论同时存在于现有集合论中两种性质上截然不同的"无穷集合"概念———潜无穷式的与实无穷式的无穷集合.指出现有数学中通常所使用的"无穷"概念极端的模糊不清,一方面混淆了原来的"有穷"概念,另一方面使人们无法理解与解析数学中与"无穷"相关的许多数量形式的性质,因此导致许多错误的数学行为.数学基础理论中这个致命缺陷是导致第二次与第三次数学危机的最根本原因.因此,只有对传统的"无穷"概念进行修正,构建新的"有穷-无穷"理论体系及相关的数量体系才是彻底解决第二次与第三次数学危机的惟一途径.
With the idea and the technique of the newly constructed basitheoryology, this article discusses two different concepts of infinite set existing in present set theory: potential infinity and actual infinity. With some examples, we discover that the general 'infinite' concept employed in mathematics today is very much vague and confused which not only unable people to understand and explain correctly the nature of some number forms related to 'infinite' but also confuses our idea on 'finite'. This fatal defect in the foundation of mathematics causes all the troubles and paradoxes related to the second and the third mathematical crises. The conclusion is: to reform the traditional 'infinite' concept and to construct a new finite-infinite theory as well as its related number spectrum is the only way to solve thoroughly the second and the third mathematical crises.
出处
《喀什师范学院学报》
2005年第3期84-87,共4页
Journal of Kashgar Teachers College
