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.
Deep Dive into Addressing mathematical inconsistency: Cantor and Godel refuted.
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
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:
-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.
-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.
-Proofs of incompleteness that use diagonal arguments (e.g. those used in Gödel’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.
2000 Mathematics Subject Classification. Primary 03B0, 03E30, 03E50; Secondary 03F07, 03F40.
. The InconsIstency of ZFC The consistency of classical set theory, more precisely the combination of the Zermelo-Fraenkel axioms and the Axiom of Choice (ZFC) [35,39,56], has recently been challenged by the construction of a proof of Goodstein’s Theorem in first-order arithmetic [55]. Goodstein’s Theorem is a statement about basic arithmetic sequences and, as such, has a purely number-theoretic character [36,56,59]. It was originally proved, not by means of first-order arithmetic, but by using the well-ordered properties of transfinite ordinals [36,56]. It was later arXiv:1002.4433v1 [math.GM] 25 Feb 2010
When Georg Cantor established the foundations of set theory, his primary objective was to provide rigorous operational tools with which to characterise the evasive and largely paradoxical concept of infinity [5,23]. He built on the work of Dedekind who, by elaborating on the observations made by Galileo and Bolzano on the nature of infinite systems [5], defined a set S as infinite when it is possible to put all its elements into one-to-one correspondence with the elements of at least one of its proper subsets; that is, to show that both set and subset have the same cardinality [5,26,27]. This property (not shared by finite sets) provided shown by Paris and Kirby to be unprovable in Peano Arithmetic (PA), with a proof that needs both transfinite induction and Gödel’s Second Incompleteness Theorem [46,59]. A previous result was required, i.e. the proof of the consistency of PA, also achieved by transfinite induction [29,30]. Consequently, if PA were capable of proving Goodstein’s Theorem, PA would also be able to prove its own consistency, thus violating Gödel’s Second Theorem [46]. Hence, the claim was made that PA cannot prove Goodstein’s Theorem. Since this claim has now been disproved [55], it is important to establish the cause of the inconsistency of ZFC.
While the problem may lie with the formulation of the axioms of ZFC, it is also possible that it lies in one or more of the basic concepts underlying the claim of Paris and Kirby [46]. This article undertakes a systematic review of these key concepts, namely transfinite number theory and mathematical incompleteness.
The article starts with a review of Cantorian mathematics (Section 2). Section 3 deals with Cantor’s proofs of nondenumerability and provides a methodical refutation of all the main rationales by showing logical inconsistencies and by identifying errors. Section 4 describes constructive arguments that prove the countable nature of the power set of the natural numbers P ( N ) and, consequently, of the set of real numbers R .
The refutation of Cantor’s proofs and the demonstration of denumerability have profound implications for many areas of mathematics. Section 5 proposes a simple and practical principle to avoid flawed implementation of the reductio method of proof. Section 6 introduces a number of conjectures, principles and definitions that may lead to the simplification of the axiomatic principles of set theory, and provide the means to compare quantitatively the size of infinite sets. Cantor’s diagonalization argument was widely adopted as a method of proof in the field of logic [20,37] and its implementation is central to important results on mathematical incompleteness by Gödel and others [3,59] . The refutation of the diagonalization argument is extended in Section 7 to the various proofs of arithmetic incompleteness. The denumerability of P ( N ) is used to construct a proof for the completeness of first-order arithmetic that confirms the above refutations and reinstates Hilbert’s Programme [4,59] .
The results reported in this article, combined with the elementary proof of Goodstein’s Theorem already cited [55],
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