Not Denumerability Of Rational Numbers

The famous Cantor’s demonstration of the Denumerability of the Rational Numbers is based on the wrong use of the term all and, in the tabular representation, on the wrong use of the limits.

In this paper it’s first rigorously showed that the Cantor demonstration is erroneous. Then two direct proofs and two indirect proofs that Rational Numbers are not-denumerable are shown.

The direct proofs are the not bijectivity between ℕ and ℚ, and the not existence of a whatsoever successor operator in ℚ. In the first indirect proof will be showed that denumerability of Rational Numbers leads to a null Lebesgue measure of any interval of ℝ. In the second indirect proof will be demonstrated that the Power Set {xⁿ}, n ∈ ℕ₀; and the Trigonometric Set {sin mx, cos nx}, m,n ∈ ℕ₀; of L² are not equivalent.