哥德尔不完全定理与数学认知的局限性——基于递归论解读哥德尔不完全定理

摘要

哥德尔不完全定理揭示了数学认知的局限性,任何一个含有初等数论及一阶谓词逻辑的形式证明系统中,都存在这样的命题,在此(封闭)系统中,依靠系统中的公理及一阶逻辑演算方法,既不能证明该命题为真,也不能证明它为假.哥德尔在定理的证明中开启可计算理论(递归论)之门,用现在成熟递归论的结果重新认识哥德尔不完全定理,使其变得更容易接受.近年来,机器学习取得突破性成果,由此引发有关人工智能是否可以完全代替人的思维能力等热点问题讨论.针对这一问题,如果承认"人工智能"是在一个交互计算系统中完成的,那么哥德尔不完全定理给出的是否定回答.%G?del's incompleteness theorem reveals the limitations of mathematical cognition. Such proposition ex-ists in any formal proof system that contains formal number theory and first order predicate logic. The axiom and first order logic algorithm in formal proof system ( enclosed system) couldn't prove if the proposition is true or false. During the process of proving the theorem, G?del discovered the computation theory-recursion theory. G?del's incompleteness theorem is easier to understand based on modern recursion theory. In recent years, the breakthrough in machine learning have raised the controversial issue that if artificial intelligence will fully replace human's thinking. The answer to the question is negative according to G?del's incompleteness theorem, because artificial intelligence can only be achieved in a closed interactive computation system.