揭穿康托尔：新的集合论与逻辑考量
Debunking Cantor: New Set-Theoretical and Logical Considerations
AI总结：
本文通过反例驳斥康托尔对角线论证，指出其逻辑缺陷，证明自然数幂集及实数集可数，并提出相对基数概念以定量比较无限集大小。
摘要：
一个多世纪以来，康托尔的超限数理论在集合论中扮演了关键角色，其影响延伸至数学的许多领域。本文以全新的视角审视康托尔理论的关键事实，扩展了先前的研究成果：i) 康托尔广为人知的对角线论证法（CDA）被一系列反例彻底驳斥，这些反例揭示了该证明的谬误。ii) 重新审视了CDA的逻辑不一致性，揭露了CDA在实施反证法时的缺陷。iii) 自然数集合的幂集P(N)的可数性通过一个充分考虑了N的所有无限子集的证明得以证实。这一结果证实了实数集R的可数性，从而也证实了连续统的可数性质。iv) 鉴于（可能）所有无限集的可数特性使得用一一对应来比较它们变得毫无意义，本文引入了一个新的相对基数概念，以便对它们的不同大小进行定量评估。
英文摘要：
For more than a century, Cantor's theory of transfinite numbers has played a pivotal role in set theory, with ramifications that extend to many areas of mathematics. This article extends earlier findings with a fresh look at the critical facts of Cantor's theory: i) Cantor's widely renowned Diagonalization Argument (CDA) is fully refuted by a set of counter-examples that expose the fallacy of this proof. ii) The logical inconsistencies of CDA are revisited, exposing the short-comings of CDA's implementation of the method of proof by contradiction. iii) The denumerability of the power set of the set of the natural numbers, P(N), is substantiated by a proof that takes full account of all the infinite subsets of N. Such a result confirms the denumerability of the set of the real numbers, R, and with it the countable nature of the continuum. iv) Given that the denumerable character of (probably) all infinite sets makes their comparison in terms of one-to-one correspondences a rather pointless exercise, a new concept of relative cardinality is introduced which facilitates a quantitative evaluation of their different magnitudes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
