无穷性的悖论与公理集合论思想述略

The Paradox of Infinite Sets and the Crisis of Mathematical Basis

康托尔首次引进无穷集合的概念,深刻揭示了无穷的本质特性,从根本上改造了数学的结构,促进了数学新分支的建立和发展。罗素悖论的出现表明集合论是有漏洞的,集合论产生悖论的根源在于集合定义中的自我指称、否定性概念以及与总体、无限的关系。公理化集合论的构建,为数学基础开辟了一个全新的平台。通过集合论的公理化,降低了悖论对数学的威胁。

Cantor introduced the concept of infinite set in the first time,and profoundly revealed the infinite essential characteristics,and also fundamentally transformed the structure of mathematics,and also promoted the establishment and development of new branches of mathematics.The emergence of Russell’s paradox shows that set theory is flawed.The root of set theory is the self-referential,negative concept and the relationship with the whole and infinity in the definition of set.The construction of axiomatic set theory opens up a new platform for the mathematical foundation.Through the axiomatization of set theory,the threat of public opinion to mathematics is reduced.