[1]HE Hua-can,HE Zhi-tao.Reunifying concepts of infinity[J].CAAI Transactions on Intelligent Systems,2010,5(3):202-220.
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Reunifying concepts of infinity

CAAI Transactions on Intelligent Systems[ISSN 1673-4785/CN 23-1538/TP] Volume: 5 Number of periods: 2010 3 Page number: 202-220 Column: 综述 Public date: 2010-06-25
Title:
Reunifying concepts of infinity
Author(s):
HE Hua-can1;  HE Zhi-tao2
1.School of Computer Science, Northwestern Polytechnical University, Xi’〖KG-*1/4〗an 710072, China;
2.School of Computer Science, Beihang University, Beijing 100191, China
Keywords:
actual infinity;  Turing machine;  infinite coding invariance;  continuum hypothesis;  Hilbert’s problems; natural number set
CLC:
O144.1; TP18
DOI:
-
Abstract:
Georg Cantor was the first person to use mathematical methods to systematically study the concept of infinity. This work formed the basis of set theory, which in turn evolved into the primary theoretical basis for modern mathematics. However, the continuum hypothesis (CH) and the layered actual infinity view in set theory introduced long lasting paradoxes into modern mathematics. Over the past 130 years, many people have worked on CH, but no one has found a way to prove or disprove it. The authors proposed a solution based on the Turing machine. The complete encoding algorithm and the complete decoding algorithm revealed the infinite coding invariance (ICI principle), proving that the real number set is denumerable and CH does not hold. In this way the reunification of the concepts of infinity was established, effectively solving Hilbert’s problem 1. Furthermore, it was proven that infinite sets can all be derived from the natural number set, and the natural number set is the mathematical model of all infinite sets. Finally, we discussed some problems with the mathematical philosophy of infinity. The establishment of a unified concept of infinity forms the basis of an actual infinity theory. It will have a significant effect on mathematics, physics, logic, philosophy and many other scientific disciplines. 
References:
［1］艾克塞尔 A D. 神秘的阿列夫［M］. 左平,译. 上海: 上海科学技术文献出版社,2008: 7482.
［2］彭漪涟,马钦荣. 逻辑学大辞典［M］. 上海: 上海辞书出版社,2004: 431,481.
［3］张顺燕. 数学的源与流［M］. 北京: 高等教育出版社, 2000: 1246.
［4］赵焕光. 数的家园［M］. 北京: 科学出版社, 2008: 7780, 115117, 315317.
［5］朱梧槚，肖奚安. 集合论导引［M］.大连: 大连理工大学出版社, 2008: 6898.
［6］DAUBEN J W. 康托的无穷的数学和哲学［M］. 郑毓信,刘晓力,译. 大连: 大连理工大学出版社, 2008: 108109.
［7］沈卫国. 论自然科学的若干基本问题［M］. 福州：海风出版社，1998: 153.
［8］沈卫国. 论实数集（连续统）的可数性及其相关问题［J］. 天津职业院校联合学报, 2006, 5: 112122.
SHEN Weiguo. Discuss about the countability of set of real numbers and the relevant problems［J］. Journal of Tianjin Vocational Institutes, 2006, 5: 112122.
［9］徐利治. 徐利治谈数学哲学［M］. 大连: 大连理工大学出版社, 2008: 173177.
［10］FATICONI T G. The mathematics of infinity: a guide to great ideas［M］. Hoboken, USA: John Wiley & Sons, Inc.〖KG-*1/4〗, 2006: 103162.
［11］MAOR E. 无穷之旅——关于无穷大的文化史［M］. 王前,武学民,金敬红,译. 上海: 上海教育出版社, 2000: 7592.
［12］朱梧檟. 数学与无穷观的逻辑基础［M］. 大连: 大连理工大学出版社, 2008: 203296.
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