Finite and infinite sets: Definitions and intuitions

The study explores the intuitive methods different-aged students use to determine whether a given set is finite or infinite; it also examines the relationship between these methods and the accepted, mathematical definition of infinite sets. It was found that two main methods were intuitively used by students: examining the finite/infinite nature of a process involving the elements of a given set, and comparing the given set with another, seemingly ‘richer’ set. The intuitive use of each of these methods largely departed from the formal one. In the concluding part it is argued that the inner contradictions in students' intuitive grasp of infinite sets could be used to increase their awareness of the fundamental role of definitions in mathematics.