对无穷集合论基本概念的几点批判

Some Criticism on the Basic Concept of Infinite Set Theory

摘要: Cantor集合论是现代数学尤其是微积分理论的基础，但其理论存在诸多逻辑漏洞，有必要明确指出，供学界批判。历史上，Cantor使用对角线法证明来实数的不可数性质，但是这种证明存在如下问题：Cantor使用无限小数进行证明具有不严密性，实数和无限小数并非等价的关系；使用十进制小数进行证明，在证明过程中会漏掉10^ℵ₀-ℵ₀个数；Cantor的证明的前提假设是“局部可以和整体进行一一对应”，这个假设未经证明因而不能说明其合理性。最后，本文在Cantor的关于“局部可以和整体一一对应”的前提下推导出了与传统集合论相反的结果，从而更加详尽的说明其荒谬性。

Abstract: Cantor’s Set theory is the basis of modern mathematics, especially calculus theory, but there are many logical loopholes in its theory. It is necessary to point out clearly for criticism. Historically, Cantor used the diagonal method to prove the uncountable nature of real numbers, but this proof has the following problems: Cantor uses infinitesimal numbers to prove which is not tight, actually real numbers and infinite decimals are not equivalent; in the process of proof with infinite decimals,10^ℵ₀-ℵ₀numbers will be missed; the premise of Cantor’s proof is that “the part can be one-to-one with the whole”. This assumption is not proved and therefore cannot explain its rationality. Finally, in this paper, Cantor’s “one-to-one correspondence with the whole” can be used to derive the opposite result from the traditional set theory, so as to explain its absurdity in more detail.