康托的超穷数理论

THEORY OF TRANSFINITE NUMBER OF CANTOR

康托（G. Cantor,1845—198）所创立的超穷集合论,在近代数学史上是令人极为惊异的巨大成就。但究其历史根源,正是由于研究分析学的基础而激起了康托对点集的兴趣,并由此而发现了超穷数.集合论,至少部分地起源于黎曼等人对三角级数的丰富研究以及对不连续函数的分析;康托对那些使函数不连续或收敛问题变得非常困难的点的集合进行了深刻的研究,并在这一过程中系统地建立和发展了一般点集的理论,从而开拓了一个全新的数学领域。本文将就其如何发现超穷数理论与创立超穷集合论的数学历史背景及其发展过程予以较系统地介述与评析。

The theory of transfinite set founded by Cantor is a great surprising achievement in the modern histroy of mathematical development. But observe and study its historicl source, it was to deal with the foundation of analysis that prompted Cantor to investigate the point set, and just then the transfinite number was discovered. Set theory, at least partly, originated from the study of trigonometric series in abundance and the analysis for discontinuous functions which were done by Ricmann ct al; Cantor investigated profoundly the set of points at which the function in question is discontinuous or the convergent problem becomes a very difficult one, and in this course he founded and developed the general point set theory systematically thus an entire new realm of mathematics was opened up. In this paper, we will rather systematically introduce and review how the theory of transfinite number was discovered and the mathematical historical background of foundation of transfinite set theory as well as its development process.