Addressing mathematical inconsistency: Cantor and Godel refuted

TL;DR

This paper critically challenges Cantor's proofs of uncountability, refutes the continuum hypothesis, and proposes that all infinite sets, including the real numbers, are countable, leading to a simplified set theory and reinstating Hilbert's Program.

Contribution

It provides a refutation of Cantor's proofs, introduces constructive proofs for the countability of the reals, and simplifies set theory by excluding transfinite numbers.

Findings

Cantor's proofs of nondenumerability are refuted due to logical inconsistencies.

Constructive proofs support the denumerability of P(N) and R.

A new theory of arithmetic is proposed that is complete, consistent, and decidable.

Abstract

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.