浅谈反证法的可操作性——基于康托尔对角线法、哥德尔不完全性定理、图灵停机问题及EPR悖论

Introduction to the operability of reduction to absurdity ——based on cantor diagonal method, Godel's incompleteness theorem, Turing downtime problems and EPR paradox

一直以来,康托尔对角线法总是与反证法密不可分,然而反证法并不如通常看得那样简单。文章从操作主义的观点,针对反证法提出了几点可操作性的要求,然后分析了几个著名的反证法论证,发现都不同程度地存在一些问题。由于不恰当的隐性假设,康托尔关于实数集不可数的证明是无效的。哥德尔为证明不完全性定理而引入的一个定理违反了矛盾律,并且他关于"可证"与"真"的区分实际上是陷入了循环论证。图灵停机问题其实是比较晚近的提法,与图灵的原始论文有较大差别,而且有些证明思路可能还或多或少地误解了图灵。最后,通过分析爱因斯坦的EPR悖论,进一步强调了假设唯一以及事实认定,对于反证法的重要性。

For a long time, cantor diagonal method and reduction to absurdity is inseparable, however, reduction to absurdity is not as simple as we usually think. According to the operating point of view, this paper puts forward some operational requirements in view of the reduction to absurdity. Then analyzed the text of several famous argument, found that there are some problems in varying degrees. Due to improper implicit assumptions, it is invalid that cantor prove real number set is uncountable. To prove incompleteness theorem, an important theorem of Godel introduced is a violation of the law of contradiction, and his excuse is a circular argument. Turing halting problem is relatively recent, but some of the popular view may be more or less misunderstood Turing. Finally, through the analysis of Einstein EPR paradox, further emphasized in reduction, assuming that the only recognized the importance of with the fact, for the reduction to absurdity, Einstein EPR paradox shows that the only assumption is very important.