A new perspective of countable and uncountable infinite sets on Georg Cantor’s definition in set theory

Abstract Georg Cantor defined countable and uncountable sets for infinite sets. Natural number set is defined as a countable set, and real number set is proven as an uncountable set by Cantor’s diagonal method. However, in this paper, natural number set will be proven as an uncountable set using Cantor’s diagonal method, and real number set will be proven as a countable set by Cantor’s definition. The process of argumentation provides us new perspectives to consider about the size of infinite sets.