潜无穷、实无穷探析

Analysis of the Potential Infinity and Actual Infinity

究竟什么是潜无穷、什么是实无穷,它们之间究竟存在着什么样的关系?无穷概念从诞生之日起就伴随着潜无穷和实无穷的长期论争。尽管不同的哲学家、逻辑学家和数学家持有不同的无穷观,但是历史上他们对于潜无穷和实无穷给出的都是直观的描述,一直没有给出严格的数学定义;对于潜无穷和实无穷之间的关系也是莫衷一是。通过对潜无穷、实无穷特征的分析,我们可以尝试给出一个严格的潜无穷和实无穷定义,利用这一定义可以对若干悖论进行新的解释,并且可以对集合论研究提供一个新的视角。

What is the potential infinity,what is the actual infinity,and what is the relation between them? There has been a long history of the contest between the potential infinity and actual infinity from the the birth of infinite concept.Although different philosophers,logicians and mathematician have different viewpoints of infinity,but they gave only the intuitional description and they didn't give the strict mathematic definition for the potential infinity and the actual infinity in history.We are unable to decide which is right for the relation between the potential infinity and the actual infinity.By analysis the character of the potential infinity and the actual infinity,we attempted to give a strict definition of the potential infinity and the actual infinity,we can use this definition to re-explain the paradox and provide a new angle of view for studying the set theory.