解决数学不一致性：康托和哥德尔反驳
Addressing mathematical inconsistency: Cantor and Godel refuted
摘要：
本文批判性地重新评价支持康托超限数理论的论点。以下结果报告：i）康托的证明不可可数性被反驳，通过分析逻辑不一致的实施reductio方法的证明，并通过识别错误。特别注意的是对角化参数和无穷大公理的解释。ii）设计了三个构造性证明，支持自然数的幂集P（N）的可数性，从而暗示了实数集R的可数性。这些结果导致了一个定理的连续性，取代康托的连续统假设，并建立了可数性质的实数线，建议所有的无限集是可数的。一些直接影响denumerability的讨论：i）有效的证明不应该包括不可想象的陈述，定义为陈述，可以被发现是假的，总是导致矛盾。这是形式化的一个原则的可设想的证明。ii）集合论的公理原理的实质性简化可以通过排除超限数来实现。为了便于比较集合，无限的以及有限的，相对基数的概念被引入。iii）使用对角线方法的不完备性证明（例如哥德尔定理中使用的证明）被反驳。基于P（N）的可数性，给出了一个构造性的证明，证明了一阶算术理论的存在性，该理论是相容的、可靠的、负完全的、可判定的，并且（假设p. r.能够证明自己的一致性。这样的结果恢复了希尔伯特纲领，并将算术完备性带到了数学的前沿。
英文摘要：
This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in implementation of the reductio method of proof and by identifying errors. Particular attention is given to the diagonalization argument and to the interpretation of the axiom of infinity. ii) Three constructive proofs have been designed that support the denumerability of the power set of the natural numbers, P(N), thus implying the denumerability of the set of the real numbers R. These results lead to a Theorem of the Continuum that supersedes Cantor's Continuum Hypothesis and establishes the countable nature of the real number line, suggesting that all infinite sets are denumerable. Some immediate implications of denumerability are discussed: i) Valid proofs should not include inconceivable statements, defined as statements that can be found to be false and always lead to contradiction. This is formalized in a Principle of Conceivable Proof. ii) Substantial simplification of the axiomatic principles of set theory can be achieved by excluding transfinite numbers. To facilitate the comparison of sets, infinite as well as finite, the concept of relative cardinality is introduced. iii) Proofs of incompleteness that use diagonal arguments (e.g. those used in Godel's Theorems) are refuted. A constructive proof, based on the denumerability of P(N), is presented to demonstrate the existence of a theory of first-order arithmetic that is consistent, sound, negation-complete, decidable and (assumed p.r. adequate) able to prove its own consistency. Such a result reinstates Hilbert's Programme and brings arithmetic completeness to the forefront of mathematics.
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