数学基础若干问题的创新性思考

Innovative Thinking on Several Basic Problems of Mathematics

摘要: 在前期一系列论文的基础上，提出康托对角线法以及区间套法在证明实数集合不可数过程中的明显逻辑问题，其本质是反证法在这个具体的使用案例中出现了以往难以察觉的问题。实数集合的不可数性尽管广为人们所接受，但其造成的问题很多。它使所谓的数学基础变得极其庞杂繁复并充满矛盾。甚至比指望其提供坚实基础的其它数学分支还要复杂混乱，这只能说明该理论本身有问题。本文的实际意义有两个方面，一是数学基础、集合论方面，提出并澄清了一系列的问题，有助于它们的健康发展；二是逻辑学方面，提示人们反证法使用中的误区，提醒人们对任何逻辑、数学结论、证明，都应该采取更为严格、慎重的态度。本文还讨论了哥德尔定理相关问题及微分法的无须极限和无穷小概念的最简理论问题。

Abstract: On the basis of previous series of papers, cantor diagonal method is put forward and the nested interval method proves that the obvious logic problems in the process of real number are uncountable. Its essence is that the use of reduction to absurdity in this specific case in the past is difficult to detect. Although the uncountability of the set of real numbers is widely accepted, it causes many problems. It makes the so-called mathematical basis extremely complex and full of contradictions. Even more complex and confusing than the other branches of mathematics that are expected to provide a solid foundation, this only suggests that the theory itself has problems. There are two aspects to the practical significance of this paper. One is the mathematical basis and set theory. Second, in the aspect of logic, people should be reminded of the mistakes in the use of proof by contradiction, and that people should adopt a more rigorous and prudent attitude towards any logical, mathematical conclusion and proof. In this paper, we also discuss the problems related to Godel’s theorem and the simplest theoretical problems about the concept of non-limit and infinitesimal of differential method.