无穷:数学分析中的一些关键概念由此分崩离析

Infinity, where some of the key concepts of analysis start to fall apart

研究实无穷和潜无穷以及它们是否相等.在构建了一个范例来证明这两个概念可以导致不同的答案之后,研究假设它们相同或不同所能够带来的影响.然后检查现代数学是如何根据需要选择性的应用这两个假设.基于讨论结果,重新审视伯克利(Berkeley)悖论和罗素(Russell)悖论,并发现前者的阴影仍然存在于现代数学体系中,而后者仅仅是一个自相矛盾的命题和谬论.

This paper studies the concepts of actual and potential infinities by addressing whether or not they are different from each other. After constructing an example that shows how these concepts can and do lead to different answers, we look at the impacts of assuming either that they are the same or that they are different. Then, we turn our attention to checking how the current state of affairs of modern mathematics unconsciously applies both of these two assumptions simultaneously depending on which one is needed to produce desired conclusions. Based on the discussions of this paper, we pay a new visit to the Berkeley and Russell′s paradoxes and find that the shadow of the former paradox still presently lingers, while the latter is nothing but simply a self-contradictory proposition and a fallacy.