摘要
从新构建的基础理论学的角度,以全新的基础理论学的方法[1],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.指出现有集合论中通常所使用的“无穷”概念极端的模糊不清,其直接的后果就是: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·集合论基础理论中这个致命缺陷是导致第三次数学危机的最根本原因·得出明确的结论:迄今为止,所有与无穷小、无穷大相关的问题与悖论在现存的含有潜无穷与实无穷概念的无穷理论体系中是不可解的,只有对现有的“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[2、3]才是构建健康的无穷集合论的唯一途径.
With the idea and the technique of the newly constructed basitheoryolog , 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： all the problems and paradoxes related to infinity and infinitesimal till now are unsolvable in the present infinite system containing the concepts of potential infinity and actual infinity, so to reform the present “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
《宜春学院学报》
2005年第4期26-29,共4页
Journal of Yichun University
关键词
基础理论学
新陈代谢
潜无穷
实无穷
无穷集合中的元素
一一对应
第三次数学危机
第二次数学危机
basitheoryology
metabolism
potential infinity
actual infinity
the elements of infinite set
one - for- one correspondence
the third mathematical crisis
the second mathematical crisis
