  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.
