The property of the set of the real numbers generated by a Gelfond-Schneider operator and the countability of all real numbers

TL;DR

This paper investigates the properties of a set of real numbers generated by a Gelfond-Schneider type exponential operator with rational arguments, revealing a contradiction related to the set's cardinality and countability.

Contribution

It introduces a new set generated by a Gelfond-Schneider operator and analyzes its cardinality, highlighting a contradiction with the known uncountability of real numbers.

Findings

The generated set has cardinality equal to the continuum.

The set is shown to be countable.

Contradiction with the uncountability of real numbers.

Abstract

Considered will be properties of the set of real numbers $\Re$ generated by an operator that has form of an exponential function of Gelfond-Schneider type with rational arguments. It will be shown that such created set has cardinal number equal to ${\aleph_0}^{\aleph_0}=c$. It will be also shown that the same set is countable. The implication of this contradiction to the countability of the set of real numbers will be discussed.