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

Considered will be properties of the set of real numbers $\mathrm{\Re }$ 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 ${{\mathrm{\aleph }}_0}^{{\mathrm{\aleph }}_0}=c$. 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.