摘要
康托尔是用数学方法系统研究实无穷概念的第一人,为此他创立了集合论,为现代数学奠定了重要的理论基础,但其中的连续统假设和层次实无穷观又给数学带来了许多问题.130多年来不断有人怀疑连续统假设,但一直没有找到解决这个问题的有效办法.文章首先在图灵机基础上提出完全编码算法和完全译码算法,揭示了无穷编码的不变性(ICI原理),证明了实数可数、连续统假设不成立,实现了实无穷概念的重新统一,从根本上解决了希尔伯特第一问题.然后进一步证明所有的无穷集都可通过自然数集变换出来,自然数集是所有无穷集的数学模型.最后讨论了有关无穷的数学哲学问题.无穷概念的统一奠定了实无穷理论的基础,对数学、物理、逻辑、哲学和其他许多学科都将产生广泛而深远的影响.
Georg Cantor was the first person to use mathematical methods to systematically study the concept of in-finity.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 intro-duced 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 estab-lishment 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.
出处
《智能系统学报》
2010年第3期202-220,共19页
CAAI Transactions on Intelligent Systems
基金
国家自然科学基金资助项目(60273087
60575034)
西北工业大学基础研究基金资助项目(W018101)
关键词
实无穷
图灵机
无穷编码的不变性
连续统假设
希尔伯特问题
自然数集
actual infinity
Turing machine
infinite coding invariance
continuum hypothesis
Hilbert’s problems
natural number set
