实数不可数之谬与超越数实数复数向量数皆可数的证明

Real Set is the Fault of Uncountable Set and the Proof that Transcendental Set, Real Set, Complex Set and Vector Set are Countable Sets

摘要: 康托用闭区间套定理和对角线方法分别证明了实数不可数，但一直以来备受争议。深刻剖析了此两种证法并揭示了其错误实质，用闭区间套定理证明实数不可数构成语形语义悖论，用对角线方法证明实数不可数犯了偷换概念、以偏概全、不当推断的逻辑错误。发现并构造了体现数学特点的数形结合、逻辑严密的方法：可能取数滚动轮排无限列表图示法、可能取数有限列表图示法、可能取数圆形数盘图示法，和拓广连分数法、代数数生成超越数法、拓广康托方程指标法，分别证明了超越数集、实数集、复数集、向量数集都是可数集。

Abstract: Cantor proves that real numbers are not countable by using closed interval nesting theorem and diagonal method, respectively. But it has always been controversial. This paper deeply analyses these two kinds of proofs and reveals the essence of their mistakes. Using the closed interval nest theorem to prove that the set of real numbers is not countable constitutes the paradox of linguistic form and semantics. Using the diagonal method to prove that the real number is uncountable makes a logical mistake of changing concepts secretly, generalizing them partially and reasoning inappropriately. We have found and constructed the method of combining numbers and graphics, which embody the characteristics of mathematics, with strict logic. An Infinite List Graphic Method for Rolling Wheel Arrangements with Possible Numbers、Graphic Method of Limited List of Possible Numbers, Diagrammatic method of circle number disc with possible number，and Extended Continuous Fraction Method、Algebraic Number Generation Transcendental Number Method、Extending the Index Method of Cantor Equation，It is proved that the transcendental set, the real set, the complex set and the vector set are all countable sets.