逻辑及数学演算中的不动项与不可判定命题(Ⅰ)

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Title:

Fixed terms and undecidable propositions in logical and mathematical calculus(Ⅰ)

作者:

张金成

中央党校函授学院, 安徽 广德 242200

Author(s):

ZHANG Jincheng

Correspondence School of the C.P.C. Central Party School, Guangde 242200, China

关键词:

正项; 反项; 不动项; 悖论; U外不动项; 不可判定命题; 不完全定理; 对角线方法; 不可数; 停机问题

Keywords:

positive term; inverse term; fixed term; paradox; fixed term outside U; undecidable proposition; incomplete theorem; diagonal method; uncountable set; halting problem

分类号:

TP18;O141

DOI:

10.3969/j.issn.1673-4785.201310076

摘要:

不动点是一个广泛而深刻的数学现象, 它已经渗透到数学的各个领域。把不动点推广到逻辑思维领域, 将证明Russel悖论是集合论中的不动项, G?del不可判定命题是自然数系统 N中的不动项, Cantor对角线方法构造的项是不动项, 不可判定的Turing机也是不动项。进一步可以证明, 当一个已知集合U可以分割成正、反集合时, 不动项不在正集或反集之中, 不动项一定是U外不动项, U外不动项的逻辑性质相对于U已经发生变异, 是未定义项, U外不动项命题是不可判定的, 这是系统的固有现象。自然数系统N中同样存在不动项, 不动项的存在与不可判定, 并不影响正、反集合的递归性与系统的完全性, 因此, G?del不完全定理的证明不成立, Cantor对角线方法证明是错误的, Turing停机问题证明也是错误的。"系统N能否完全"?实数是否可数?Turing停机问题是否可判定?都必须重新思考。

Abstract:

As a kind of broad and deep mathematical phenomenon, the fixed point has penetrated into all fields of mathematics. In this paper, the fixed point is extended to the logic thinking area and is about to prove that Russell’s paradox is the fixed term that is in accordance with the set theory and that G?del’s undecidable proposition is the fixed term within the natural number system N, a term formed by Cantor’s diagonal method is the fixed term and undecidable Turning is also a fixed term. Furthermore it can be shown that when a known set U can be divided into a positive set and an inverse set and if the fixed term is neither in the positive set nor in the inverse set, then this fixed term must be that outsid U. Thus it is an inherent phenomenon of the system that the logic property of the fixed term excluded from U has changed relative to U and the theorem of the fixed term outside U is undecidable. In addition, there are also fixed terms in the natural number system N, where the existence and undecidability does not have an effect on the recursive nature of the positive and inverse sets as well as the completeness of the system. Therefore, the mathematical proof for G?del’s theorem cannot be true and Cantor’s diagonal method is proved to be false and Turning’s halting problem is also proved to be false. Whether or not the system N can be complete, the real number is countable or not, or whether or not Turning’s halt problem can be decided should be reconsidered.