[1]ZHANG Jincheng.Fixed terms and undecidable propositions in logical and mathematical calculus(Ⅰ)[J].CAAI Transactions on Intelligent Systems,2014,9(4):499-510.[doi:10.3969/j.issn.1673-4785.201310076]
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Fixed terms and undecidable propositions in logical and mathematical calculus(Ⅰ)

CAAI Transactions on Intelligent Systems[ISSN 1673-4785/CN 23-1538/TP] Volume: 9 Number of periods: 2014 4 Page number: 499-510 Column: 学术论文—人工智能基础 Public date: 2014-08-25
Title:
Fixed terms and undecidable propositions in logical and mathematical calculus(Ⅰ)
Author(s):
ZHANG Jincheng
Correspondence School of the C.P.C. Central Party School, Guangde 242200, China
Keywords:
positive term; inverse term; fixed term; paradox; fixed term outside U; undecidable proposition; incomplete theorem; diagonal method; uncountable set; halting problem
CLC:
TP18;O141
DOI:
10.3969/j.issn.1673-4785.201310076
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.
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