Lower and upper estimates of the quantity of algebraic numbers

Abstract

It is well known that the set of algebraic numbers (let us call it $A$) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using \G1-based infinite numbers is applied to measure the set $A$ (where the number \G1 is called \emph{grossone}). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable or it has the cardinality of the continuum, the \G1-based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of $A$ are obtained. Both estimates are expressed in \G1-based numbers.