PROPOSED IS A COUNTEREXAMPLEFOR THE DIAGONAL METHOD OF GEORG CANTOR

Abstract

The diagonal method proposed by Georg Cantor is a proof that unenumerable sets exist. For example, the set of all subsets of the set of all natural numbers; or the set of all binary sequencies; or the set of all decimals fractions belong in gto the interval [0; 1) are allegedly not countable. This raises doubts about the axiom of choice.

For the set of all binary sequences, it is proposed to call “native” the enumeration defined by the rule: “If the reisabinary sequence (a₀, a₁, a₂, a₃, …), all which elements belong to set {0,1}, then its exact position in the “native enumeration” calculates by the formula a₀× 2⁰+a₁× 2¹+a₂× 2²+a₃× 2³+…, where×–the multiplication sign (all arithmetic operations are performed in one of number system, such as decimal, and without limitation of digit capacity)”.

Proofs are given that "native enumeration" is a counterexample for George Cantor's diagonal method.

Keywords: discrete mathematics, counter example, enumeration of set, enuberable set, unenuberable set, fabulist Aesop, Guilherme Figueredo, mathematical intuition, diagonal method of Georg Cantor, circus trick, infinite set, cardinality of set, the set of all subsets of a set, the set of all binary sequencies, axiom of choice, set of all real numbers, 1+2+3+4+… .