康托尔集的产生和影响  

The Emergence and Influence of the Cantor Set

作　　者：江南[1,2] 唐泉[2] 刘建新[3] JIANG Nan;TANG Quan;LIU Jianxin(School of Science,Xian Shiyou University,Xi'an 710065,China;Institute for Advanced Studies in the History of Science,Northwest University,Xi'an 710127,China;School of Education,Xinyang Normal University,Xinyang 464000,China)

机构地区：[1]西安石油大学理学院,西安710065 [2]西北大学科学史高等研究院,西安710127 [3]信阳师范大学教育学院,信阳464000

出　　处：《中国科技史杂志》2024年第1期88-98,共11页The Chinese Journal for the History of Science and Technology

基　　金：陕西省社会科学基金年度项目“分形理论及其非线性思想研究”(项目编号:2023C004);中国博士后科学基金资助项目“分形几何的历史演变探究”(项目编号:2020M683693XB)。

摘　　要：康托尔集作为分形几何早期经典的集合,溯源它的产生过程对于完善分形几何的历史有着重要的意义。在点集论萌芽和实数理论严格化的历史背景下,史密斯以处理函数的可积性问题为动机,以探究不连续函数的积分条件为出发点,以寻找不满足黎曼积分理论的具体反例为突破口,构造了外容度为正值的无处稠密集,即康托尔集的初始模型。康托尔则另辟蹊径,从研究导集的概念入手,以澄清点集的稠密性和完备性的内在关系为导向,以探寻具体的实例为目标,运用无穷级数表示理论,构造了一个完备但处处不稠密的点集。康托尔的工作相比史密斯更顺应当时的数学发展潮流,因此获得该点集的冠名权。此外,魔鬼的阶梯、分数维数、康托尔尘等分形对象在康托尔集的影响下相继产生,它们为分形几何的最终创立提供了思想源泉。The Cantor set is a typical set in the early stage of fractal geometry.It is of great significance to trace the origin of the Cantor set for perfecting the history of fractal geometry.Under the historical background of the sprout of set theory and the strictness of real number theory,Smith constructed the original model of the Cantor set which has a positive external capacity by the motivation of dealing with the integrability of functions,the starting point is to explore the integration conditions of discontinuous functions,and the breakthrough is to find specific counter examples that do not satisfy the Riemann integral theory.On the other hand,Cantor constructed a complete but nowhere dense point set by using the representation theory of infinite series.This was due to studying the concept of derived sets as the starting point,and clarifying the density and completeness of point sets as the guidance,and exploring the specific example as the goal.Cantor's work was more in line with the trend of the mathematical development at that time than Smith's,so he won the naming right for this set.In addition,the devil's ladder,fractional dimension,Cantor dust and other fractal research objects appeared one after another under the Cantor set's influence,which lay the ideological source for the final establishment of fractal geometry.

关 键 词：康托尔集 稠密性 完备性 分形几何 

分 类 号：N09[自然科学总论—科学技术哲学] O18[理学—数学]