A Cantor Trio: Denumerability, the Reals, and the Real Algebraic Numbers

Abstract

We present a formalization in ACL2(r) of three proofs originally done by Cantor. The first two are different proofs of the non-denumerability of the reals. The first, which was described by Cantor in 1874, relies on the completeness of the real numbers, in the form that any infinite chain of closed, bounded intervals has a non-empty intersection. The second proof uses Cantor’s celebrated diagonalization argument, which did not appear until 1891. The third proof is of the existence of real transcendental (i.e., non-algebraic) numbers. It also appeared in Cantor’s 1874 paper, as a corollary to the non-denumerability of the reals. What Cantor ingeniously showed is that the algebraic numbers are denumerable, so every open interval must contain at least one transcendental number.