[1]ZHANG Jincheng.Fixed terms and undecidable propositions in logical and mathematical calculus (Ⅱ)[J].CAAI Transactions on Intelligent Systems,2014,9(5):618-631.[doi:10.3969/j.issn.1673-4785.201310076]
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Fixed terms and undecidable propositions in logical and mathematical calculus (Ⅱ)

CAAI Transactions on Intelligent Systems[ISSN 1673-4785/CN 23-1538/TP] Volume: 9 Number of periods: 2014 5 Page number: 618-631 Column: 学术论文—人工智能基础 Public date: 2014-10-25
Title:
Fixed terms and undecidable propositions in logical and mathematical calculus (Ⅱ)
Author(s):
ZHANG Jincheng
Correspondence School, Communist Party College, 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:
B813;TP18
DOI:
10.3969/j.issn.1673-4785.201310076
Abstract:
As a kind of broad and deep mathematical phenomenon, fixed point has penetrated into all fields of mathematics. This paper extends the fixed point to the logical thinking. It proves that Russell’s paradox is the fixed term in accordance with the set theory. It also proves that G?del’s undecidable proposition is the fixed term within the natural number system N. The term formed by Cantor’s diagonal method is fixed term and undecidable Turning is also fixed term. Furthermore, it can be proven that when a known set U is 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 outside U. Thus, it is an inherent phenomenon of the system that the logical property of the fixed term excluded from U has changed relative to U and the theorem of fixed term outside U is undecidable. In addition, there are also fixed terms in the natural number system N, where the existence and undecidability do not exert effect on the recursive nature of positive and inverse sets and the completeness of 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 proved to be false. Whether the system N can be complete, real number is countable or not, whether Turning’s halt problem can be decided should be reconsidered.
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